Explainer: What are logarithms and exponents?

 

Logarithms are mathematical relationships used to compare things that can vary dramatically in scale.

When COVID-19 hit the United States, the numbers just seemed to explode. First, there were only one or two cases. Then there were 10. Then 100. Then thousands and then hundreds of thousands. Increases like this are hard to understand. But exponents and logarithms can help make sense of those dramatic increases.

Scientists often describe trends that increase very dramatically as being exponential. It means that things don’t increase (or decrease) at a steady pace or rate. It means the rate changes at some increasing pace.

An example is the decibel scale, which measures sound pressure level. It is one way to describe the strength of a sound wave. It’s not quite the same thing as loudness, in terms of human hearing, but it’s close. For every 10 decibel increase, the sound pressure increases 10 times. So a 20 decibel sound has not twice the sound pressure of 10 decibels, but 10 times that level. And the sound pressure level of a 50 decibel noise is 10,000 times greater than a 10-decibel whisper (because you’ve multiplied 10 x 10 x 10 x 10).

An exponent is a number that tells you how many times to multiply some base number by itself. In that example above, the base is 10. So using exponents, you could say that 50 decibels is 104 times as loud as 10 decibels. Exponents are shown as a superscript — a little number to the upper right of the base number. And that little 4 means you’re to multiply 10 times itself four times. Again, it’s 10 x 10 x 10 x 10 (or 10,000).

Logarithms are the inverse of exponents. A logarithm (or log) is the mathematical expression used to answer the question: How many times must one “base” number be multiplied by itself to get some other particular number?

For instance, how many times must a base of 10 be multiplied by itself to get 1,000? The answer is 3 (1,000 = 10 × 10 × 10). So the logarithm base 10 of 1,000 is 3. It’s written using a subscript (small number) to the lower right of the base number. So the statement would be log10(1,000) = 3.

At first, the idea of a logarithm might seem unfamiliar. But you probably already think logarithmically about numbers. You just don’t realize it.

Let’s think about how many digits a number has. The number 100 is 10 times as big as the number 10, but it only has one more digit. The number 1,000,000 is 100,000 times as big as 10, but it only has five more digits. The number of digits a number has grows logarithmically. And thinking about numbers also shows why logarithms can be useful for displaying data. Can you imagine if every time you wrote the number 1,000,000 you had to write down a million tally marks? You’d be there all week! But the “place value system” we use allows us to write down numbers in a much more efficient way.

Why describe things as logs and exponents?

Log scales can be useful because some types of human perception are logarithmic. In the case of sound, we perceive a conversation in a noisy room (60 dB) to be just a bit louder than a conversation in a quiet room (50 dB). Yet the sound pressure level of voices in the noisy room might be 10 times higher.

These graphs plot the same information, but show it somewhat differently. The plot at left is linear, the one at right is logarithmic. The steep curve in the left plot looks flatter on the right plot.CANADIAN JOURNAL OF POLITICAL SCIENCE, APR. 14, 2020, PP.1–6/ (CC BY 4.0)

Another reason to use a log scale is that it allows scientists to show data easily. It would be hard to fit the 10 million lines on a sheet of graph paper that would be needed to plot the differences from a quiet whisper (30 decibels) to the sound of a jackhammer (100 decibels). But they’ll easily fit on a page using a scale that’s logarithmic. It’s also an easy way to see and understand big changes such as rates of growth (for a puppy, a tree or a country’s economy). Any time you see the phrase “order of magnitude,” you’re seeing a reference to a logarithm.

Logarithms have many uses in science. pH — the measure of how acidic or basic a solution is — is logarithmic. So is the Richter scale for measuring earthquake strength.

In 2020, the term logarithmic became best known to the public for its use in describing the spread of the new pandemic corona virus (SARS-CoV-2). As long as each person who got infected spread the virus to no more than one other person, the size of the infection would stay the same or die out. But if the number was more than 1, it would increase “exponentially” — which means that a logarithmic scale could be useful to graph it.

Basic bases

The base number of a logarithm can be almost any number. But there are three bases which are especially common for science and other uses.

1. Binary logarithm: This is a logarithm where the base number is two. Binary logarithms are the basis for the binary numeral system, which allows people to count using only the numbers zero and one. Binary logarithms are important in computer science. They’re also used in music theory. A binary logarithm describes the number of octaves between two musical notes.
2. Natural logarithm: A so-called “natural” logarithm — written ln — is used in many areas of math and science. Here the base number is an irrational number referred to as e, or Euler’s number. (The mathematician Leonhard Euler did not intend to name it after himself. He was writing a math paper using letters to represent numbers and happened to use e for this number.) That e is about 2.72 (though you can never write it down completely in decimals). The number e has some very special mathematical properties that make it useful in many areas of math and science, including chemistry, economics (the study of wealth) and statistics. Researchers also have used the natural logarithm to define the curve that describes how a dog’s age relates to a human one.
3. Common logarithm: This is a logarithm where the base number is 10. This is the logarithm used in measurements for sound, pH, electricity and light.

How do we prioritize what we see?

 It is known that different regions of the brain help us prioritize information so we can efficiently process visual scenes. A new study by a team of neuroscientists has discovered that one specific region, the occipital cortex, plays a causal role in piloting our attention to manage the intake of images

"By briefly disrupting cortical excitability of the occipital cortex with TMS we could extinguish the known effects of involuntary, or exogenous, covert spatial attention, and thus reveal a causal link between the occipital cortex and the effect of covert attention on vision," explains Marisa Carrasco, a professor of psychology and neural science at New York University and the senior author of the paper.

"This is a surprising finding as most previous research shows that other areas of the brain -- the frontal and parietal cortex -- help us in selectively processing many images that come our way, but this research reveals that the occipital cortex also plays a critical functional role," adds Antonio Fernández, an NYU doctoral student and first author of the paper.

In our daily lives, we are bombarded with an overwhelming amount of sensory information, notably visuals, from as big as skyscrapers to as small as computer screens. In spite of this, we have the impression of effortlessly understanding what we see, unaware of the complex mechanisms that, in a kind of cognitive triage, help us prioritize the information that we process. It's been long shown that the processing of visual information and its accompanying neural computations consume a great deal of energy, which is finite and must be managed.

One of the ways we achieve this is through covert spatial attention, which enables us to select a certain location of a visual scene and prioritize its processing and guide behavior, even without moving our eyes to that location (which is why it is called covert).

Covert attention, whether voluntary (endogenous) or involuntary (exogenous), is a trade-off process -- it benefits visual processing at the attended location at the expense of processing elsewhere.

Earlier neuroimaging and electrophysiological studies have shown that visual areas in the occipital cortex, located in the back of the brain, are part of the attention cortical networks, but it was unknown whether this region is necessary in the prioritizing of visual content.

Because of its well-established role in vision, Fernández and Carrasco specifically sought to determine if the occipital cortex played a causal role in guiding involuntary (exogenous) covert attention.

To do so, they conducted a series of experiments with human observers and used TMS to manipulate and briefly alter cortical excitability in the occipital area.

The authors asked the participants to make an orientation judgement by determining if an image was tilted right or left on a computer screen. They also manipulated participants' covert attention with an additional image -- a cue (small line) that appeared on the screen prior to stimuli presentation to automatically attract attention to its location. One stimulus appeared left and the other stimulus appeared right off center, while observers fixated at a central point. The cortical representation of one of the two stimuli was briefly disrupted using TMS. In some trials, "valid trials," the cue indicated the stimulus location observers should respond to; in other trials, "invalid trials," the peripheral cue indicated the other stimulus location. In neutral trials, both stimuli were cued.

When you're smiling, the whole world really does smile with you

From Sinatra to Katy Perry, celebrities have long sung about the power of a smile -- how it picks you up, changes your outlook, and generally makes you feel better. But is it all smoke and mirrors, or is there a scientific backing to the claim?

Groundbreaking research from the University of South Australia confirms that the act of smiling can trick your mind into being more positive, simply by moving your facial muscles.

With the world in crisis amid COVID-19, and alarming rises of anxiety and depression in Australia and around the world, the findings could not be more timely.

The study, published in Experimental Psychology, evaluated the impact of a covert smile on perception of face and body expressions. In both scenarios, a smile was induced by participants holding a pen between their teeth, forcing their facial muscles to replicate the movement of a smile.

The research found that facial muscular activity not only alters the recognition of facial expressions but also body expressions, with both generating more positive emotions.

Lead researcher and human and artificial cognition expert, UniSA's Dr Fernando Marmolejo-Ramos says the finding has important insights for mental health.

"When your muscles say you're happy, you're more likely to see the world around you in a positive way," Dr Marmolejo-Ramos says.

"In our research we found that when you forcefully practise smiling, it stimulates the amygdala -- the emotional centre of the brain -- which releases neurotransmitters to encourage an emotionally positive state.

"For mental health, this has interesting implications. If we can trick the brain into perceiving stimuli as 'happy', then we can potentially use this mechanism to help boost mental health."

The study replicated findings from the 'covert' smile experiment by evaluating how people interpret a range of facial expressions (spanning frowns to smiles) using the pen-in-teeth mechanism; it then extended this using point-light motion images (spanning sad walking videos to happy walking videos) as the visual stimuli.

Dr Marmolejo-Ramos says there is a strong link between action and perception.

"In a nutshell, perceptual and motor systems are intertwined when we emotionally process stimuli," Dr Marmolejo-Ramos says.

"A 'fake it 'til you make it' approach could have more credit than we expect."

Dolphins can learn from their peers how to use shells as tools

Some bottlenose dolphins in Australia’s Shark Bay use shells to collect lunch. A dolphin will trap underwater prey in large sea snail shells. It then carries the shell to the surface and shakes the contents into its mouth.  

 Dolphins often learn how to hunt from their mothers. But Indo-Pacific bottlenose dolphins in Western Australia’s Shark Bay are different. Some may pick up one clever foraging behavior from their peers.

Previous studies had suggested dolphins can learn from peers. But the new report is the first to quantify the importance of social networks over other factors, says Sonja Wild. She’s a behavioral ecologist at the University of Konstanz in Germany.

Cetaceans include dolphins, whales and porpoises. They are known for using clever tactics to round up meals. Humpback whales off Alaska sometimes use their fins and circular bubble nets to catch fish. Indo-Pacific bottlenose dolphins in Shark Bay employ another tactic. They use sea sponges to protect their beaks while rooting for food on the seafloor. The animals learn the sponge trick from their moms.

Some Shark Bay dolphins also use a more unusual tool-based foraging method. It’s known as shelling. A dolphin will first trap underwater prey in the large shell of a sea snail. Then the dolphin pokes its beak into the shell’s opening. It can now lift the shell above the water’s surface to shake the contents into its mouth.

“It is pretty mind-blowing,” says Wild. She studied these dolphins while a graduate student at the University of Leeds in England. This brief behavior appears to be rare. From 2007 to 2018, Wild and her colleagues documented 5,278 dolphin group encounters in the western gulf of Shark Bay. There were only 42 shelling events. Only 19 dolphins exhibited the behavior.

The researchers then analyzed the behavior of 310 dolphins that had been seen at least 11 times. These included 15 shellers. They mapped the dolphins’ network of social interactions. Those interactions explained shelling’s spread better than other factors. This included genetic relatedness and the amount of environmental overlap between dolphins.

Wild likens the learning of this behavior to the spread of a virus. “Just by spending time with each other, [dolphins] are more likely to transmit those behaviors,” she says. The researchers estimate that 57 percent of the dolphins learned shelling from peers rather than on their own.

But the researchers may be too quick to dismiss the influence of the dolphins’ environment and what they learn from mom, says Janet Mann. She’s a biologist at Georgetown University in Washington, D.C. She, too, studies dolphin behavior at Shark Bay.

“Those shells are found in particular habitats. And animals who overlap in those habitats would have access to those shells.” She adds that they would “also bump into each other more often.” A dolphin’s shelling behavior could also have been influenced during the long time the animal spent with its mother.

No question, she says, “Dolphins are smart: They watch each other and see what others do.”

Scientists Say: Deforestation This is the act of removing trees from a large area

Deforestation is the cutting down of trees from large areas of land — trees that are critical to keeping our planet cool.

 

Deforestation (noun, “DEE-for-es-STAY-shun”)

This is the act of cutting down trees across a wide area. Some people also use words like clearance, clearing or clearcutting for deforestation. Forests cover a third of the world’s land. But humans cut down 26 million hectares (64 million acres) of those forests every year. That’s an area the size of the United Kingdom, or just larger than the state of Oregon.

Deforestation removes the trees in an area so the land can be used for something that’s not a forest. That can be farms, homes or businesses. Ranches in the Amazon, for example, get their space from deforestation. 

Forests take in carbon dioxide from the atmosphere. That’s especially important because people are producing so much of it. Carbon dioxide is a greenhouse gas that works like a blanket to keep in heat. Trees have been able to remove of some of our carbon dioxide. But as deforestation continues, they take in less and less.

In a sentence

Too much deforestation means that tropical forests are now producing more greenhouse gases than they absorb.

Explainer: What are Antibodies? The body makes this chemical ammo to fight foreign invaders — now and later

An artist’s illustration of antibodies (blue-green structures) attempting to latch onto the antigens on the outer surface of a coronavirus (red). Antibodies are one of the major players in the immune system’s attack against viruses.

 A world of germs is vying to invade your body and make you sick. Luckily, your immune system can assemble a mighty army to protect you. Think of this system as your own personal team of superheroes. They are dedicated to keeping you safe.

And antibodies are among their strongest ammunition. Also called immunoglobulins (Ih-mue-noh-GLOB-you-linz), or Ig’s, these are a family of proteins.

The job of these antibodies is to locate and attack “foreign” proteins — that is, proteins that don’t appear to belong in the body.

These foreign invaders contain substances the body doesn’t recognize. Known as antigens, these can be parts of bacteria, viruses or other microbes. Pollen and other things that cause allergies can have antigens, too. If someone is given blood that doesn’t match their blood type — during surgery, for instance — those blood cells can host antigens.