Evaluating Logarithms

Learning Outcomes

• Evaluate logarithms with and without a calculator.
• Evaluate logarithms with base 10 and base e.

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider ${\mathrm{log}}_{2}8$. We ask, “To what exponent must 2 be raised in order to get 8?” Because we already know ${2}^{3}=8$, it follows that ${\mathrm{log}}_{2}8=3$.

Now consider solving ${\mathrm{log}}_{7}49$ and ${\mathrm{log}}_{3}27$ mentally.

• We ask, “To what exponent must 7 be raised in order to get 49?” We know ${7}^{2}=49$. Therefore, ${\mathrm{log}}_{7}49=2$.
• We ask, “To what exponent must 3 be raised in order to get 27?” We know ${3}^{3}=27$. Therefore, ${\mathrm{log}}_{3}27=3$.

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate ${\mathrm{log}}_{\frac{2}{3}}\frac{4}{9}$ mentally.

• We ask, “To what exponent must $\frac{2}{3}$ be raised in order to get $\frac{4}{9}$? ” We know ${2}^{2}=4$ and ${3}^{2}=9$, so ${\left(\frac{2}{3}\right)}^{2}=\frac{4}{9}$. Therefore, ${\mathrm{log}}_{\frac{2}{3}}\left(\frac{4}{9}\right)=2$.

How To: Given a logarithm of the form $y={\mathrm{log}}_{b}\left(x\right)$, evaluate it mentally

1. Rewrite the argument x as a power of b: ${b}^{y}=x$.
2. Use previous knowledge of powers of b to identify y by asking, “To what exponent should b be raised in order to get x?”

Example: Solving Logarithms Mentally

Solve $y={\mathrm{log}}_{4}\left(64\right)$ without using a calculator.

Try It

Solve $y={\mathrm{log}}_{121}\left(11\right)$ without using a calculator.

Example: Evaluating the Logarithm of a Reciprocal

Evaluate $y={\mathrm{log}}_{3}\left(\frac{1}{27}\right)$ without using a calculator.

Try It

Evaluate $y={\mathrm{log}}_{2}\left(\frac{1}{32}\right)$ without using a calculator.

Using Natural Logarithms

The most frequently used base for logarithms is e. Base e logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base e logarithm, ${\mathrm{log}}_{e}\left(x\right)$, has its own notation, $\mathrm{ln}\left(x\right)$.

Most values of $\mathrm{ln}\left(x\right)$ can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, $\mathrm{ln}1=0$. For other natural logarithms, we can use the $\mathrm{ln}$ key that can be found on most scientific calculators. We can also find the natural logarithm of any power of e using the inverse property of logarithms.

A General Note: Definition of the Natural Logarithm

A natural logarithm is a logarithm with base e. We write ${\mathrm{log}}_{e}\left(x\right)$ simply as $\mathrm{ln}\left(x\right)$. The natural logarithm of a positive number x satisfies the following definition:

For $x>0$, $y=\mathrm{ln}\left(x\right)\text{ is equal to }{e}^{y}=x$
We read $\mathrm{ln}\left(x\right)$ as, “the logarithm with base e of x” or “the natural logarithm of x.”

The logarithm y is the exponent to which e must be raised to get x.

Since the functions $y=e^{x}$ and $y=\mathrm{ln}\left(x\right)$ are inverse functions, $\mathrm{ln}\left({e}^{x}\right)=x$ for all x and $e^{\mathrm{ln}\left(x\right)}=x$ for $x>0$.

How To: Given a natural logarithm Of the form $y=\mathrm{ln}\left(x\right)$, evaluate it using a calculator+

1. Press [LN].
2. Enter the value given for x, followed by [ ) ].
3. Press [ENTER].

Example: Evaluating a Natural Logarithm Using a Calculator

Evaluate $y=\mathrm{ln}\left(500\right)$ to four decimal places using a calculator.

Try It

Evaluate $\mathrm{ln}\left(-500\right)$.

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