Logarithms

When you were in elementary school, you learned that addition and subtraction were inverse operations and that multiplication and division were inverse operations. You also studied the basics of exponents. Did you know that exponents also have an inverse operation? They're called logarithms ( $\log$ for short) and answer the question "To what power do I have to raise b to get this number?". Expressed mathematically, we can say that $\log_{b} x = a$ means $b^{x} = a$ .

This article will explore what this notation means and how logarithms might be presented to you on a math test. Let's get started!

Examples of logarithms

When looking at logarithms, it's important to remember that the base of a logarithmic equation (the small number in the subscript) will also be the base of the corresponding power equation (the number raised to some power). They both stay on the left side of the equation. For example, $\log_{7} 49 = 2$ because $7^{2} = 49$ .

If we were looking at an expression such as $\log_{2} 49 = 2$ , the value is 5 because $2^{5} = 32$ .

The same principle is true if you're looking at negative exponents. For instance, $\log_{10} 0.01 = - 2$ because $10^{- 2} = 0.01$ .

This can be fairly complex, so don't worry if you don't grasp it right away. Realistically, only two bases are used frequently.

Practical applications of logarithms

In real life, the only logarithms you're likely to see are $\log_{10}$ and $\log_{e}$ . ${Log}_{10}$ is simply a logarithm in base 10 and is sometimes called the common logarithm. In fact, the common logarithm is so common that if we want $\log_{10} x$ we usually just write $\log x$ , with the base of 10 assumed. The "log" key on your graphing calculator probably defaults to base 10 as well. Of course, this means that your teacher will assume a base of 10 if you don't include a subscript. If you wanted a different base, you're in trouble.

If you want to use a different base on your graphing calculator, you can use the following change of base formula:

$\log_{b} x = \frac{\log_{10} x}{\log_{10} b}$

Log_e refers to the mathematical constant e, equivalent to roughly 2.71828183. Since e is an irrational number, we cannot possibly list all of its digits. Anyway, this logarithm is called the natural logarithm and is generally written as ln. Put another way, $\log_{e} x$ is the same thing as $\ln x$ . Most graphing calculators have an ln key for working with natural logarithms, and you can again change the base by using the change of base formula:

$\log_{b} x = \frac{\ln x}{\ln b}$

The key to logarithms is to take your time. Be sure to write legibly and type numbers into your calculator carefully where applicable. Likewise, it never hurts to check if the corresponding exponent gives you the answer the logarithm expects. For more information on how to work with logarithms, see simplifying logarithmic expressions.

Practice Questions

Logarithms practice questions

a. Evaluate log base 5 of 25

This question is effectively asking us, "what power of 5 equals 25?". We know that $5 * 5 = 25$ , meaning that $5^{2} = 25$ . The answer is 2.

b. Evaluate log base 3 of 27

This question is effectively asking us, "what power of 3 equals 27?". We know that $3 * 3 * 3 = 27$ , meaning that $3^{3} = 27$ . The answer is 3.

c. Solve for x: $\log_{x} 36 = 2$

We can rewrite the equation as $x^{2} = 36$ since $\log_{b} x = a$ means $b^{x} = a$ . Next, we solve for x just like any other equation. Taking the square root of both sides gives an answer of $x = 6$ . The math checks out as $6^{2} = 36$ .

d. What is the value of $\log_{2} 65$ ?

We need to use our graphing calculators for this one. It likely doesn't have a $\log_{2}$ key, so we use the Change of Base Formula to translate it into $\frac{\log_{10} (65)}{\log_{10} (2)}$ . Punch that equation into your calculator, and you'll find that the answer is approximately 6.0223.

e. What is the value of log base 4.5 of 381? Round your answer to the nearest hundredth.

We need to use our graphing calculators for this one as well. It definitely doesn't have a $\log_{4.5}$ key, so we use the Change of Base Formula to translate it into $\frac{\ln (381)}{\ln (4.5)}$ . Punch that equation into your calculator, and you should get an answer of 3.95112604435386.. That's a lot of decimals, but the question asks us to round to the nearest hundredth. Therefore, our final answer is 3.95.

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Get expert assistance with logarithms with Varsity Tutors

Logarithms force students to think in terms of powers and exponents, requiring a paradigm shift in how you approach mathematics. Furthermore, the topic makes extensive use of graphing calculators with a lot of buttons and a clunky interface. If you or your student need help to understand what logarithms are or how to input problems into your calculator, an experienced math tutor could provide extra practice problems until you feel comfortable moving on. You could also review step-by-step tutorials on how to approach specific problem types. Reach out to the Educational Directors at Varsity Tutors for more info.