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# Evaluating Logarithms

Concept

## Logarithm

A logarithm is the inverse of an exponential function. The logarithm of a positive number $m$ can be written as follows. The expression $\log_{b}(m)$ is read as $\log$ base $b$ of $m$ and states that raising $b$ to the $n^{\text{th}}$ power yields $m.$

$\log_{b}(m)=n \quad \Leftrightarrow \quad b^n=m$

Here, $b$ is both the base of the logarithm and the base of the exponent. For example, the value of $n$ in $\log_{4}(16)=n$ is given by the exponent to which $4$ would would be raised to result in $16.$ \begin{aligned} \log_{4}(16)={\color{#0000FF}{n}}\quad & \Leftrightarrow\quad 4^{\color{#0000FF}{n}}=16\\ \log_{4}(16)={\color{#0000FF}{2}}\quad & \Leftrightarrow\quad 4^{\color{#0000FF}{2}}=16 \end{aligned} The logarithmic form and exponential form are equivalent.

Concept

## Common Logarithm

A common logarithm is a logarithm of base $10.$ For example, $\log_{10}(1000)$ is equal to $3$ because $10^3$ is equal to $1000.$

Since $\log_{10}$ is used so often, it is sometimes written without a base. For positive values of $m,$ the common $\log$ of $m$ can be defined as follows.

$\log(m)=n \quad \Leftrightarrow \quad 10^n=m$

Concept

## The Natural Logarithm

A natural logarithm is a logarithm with base $e$. This means that $\ln(m)$ equals the exponent to which $e$ must be raised to equal $m.$

$\ln(m)=n \quad \Leftrightarrow \quad m = e^n$

Although it is correct to write $\log_{e},$ the natural logarithm is more commonly written as $\ln.$

$\begin{gathered} \ln(m) = \log_{e}(m) \end{gathered}$
Exercise

For the following expressions, rewrite them in either logarithmic or exponential form. $\log(100)=2 \quad 8^2=64 \quad \log_2(32)=5 \quad e^0=1$

Solution

We will rewrite each expression one at a time using the relationship $\log_{b}(m)=n \quad \Leftrightarrow \quad b^n=m.$ Notice that the first and third are written as logarithms and the second and fourth are written as exponents. Since a base is not written on the first logarithm, we know it's base $10.$ Furthermore, $2$ is the exponent. Thus, $\log(100)=2 \quad\Leftrightarrow\quad 10^2=100.$ The same reasoning applies for the third expression, $\log_{2}(32)=5.$ $\log_{2}(32)=5 \quad \Leftrightarrow \quad 2^5=32$ Using the same relationship in the opposite way, we can rewrite the second and fourth expression. In $8^2=64,$ it can be seen that $8$ is the base and $2$ is the exponent. This means that $64$ is the value of which we take the logarithm. $8^2=64 \quad \Leftrightarrow \quad \log_{8}(64)=2.$ For the last expression, this yields $e^0=1 \quad \Leftrightarrow \quad \log_{e}(1)=0 \quad \Leftrightarrow \quad \ln(1)=0.$ To summarize, the following expressions are equivalent. \begin{aligned} \log(100)=2 &\quad\Leftrightarrow\quad 10^2=100\\ 8^2=64 &\quad\Leftrightarrow\quad \log_8(64)=2\\ \log_2(32)=5 &\quad\Leftrightarrow\quad 2^5=32\\ e^0=1 &\quad\Leftrightarrow\quad \ln(1)=0 \end{aligned}

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Exercise

Evaluate the following logarithms. $\log_{3}(81) \quad \ln(e) \quad \log \left(\frac{1}{1000} \right)$

Solution

When evaluating logarithms, it can be helpful to think about what the expression means. $\log_{3}(81)$ asks the exponent to which $3$ must be raised to equal $81.$ Since, $3 \cdot 3 \cdot 3 \cdot 3 =81,$ $\log_{3}(81)=4.$ The second expression, $\ln(e),$ is a logarithm of base $e.$ It asks the exponent to which $e$ must be raised to equal $e.$ Since $e^1=e,$ $\ln(e)=1.$ Notice that, unlike the other expressions, the last, which is of base $10,$ contains a fraction. Thus, we must consider the exponent to which $10$ is raised to equal $\frac{1}{1000}.$ As it turns out, negative exponents yield fractions. $x^{\text{-} n} = \dfrac{1}{x^n}.$ It can be helpful to rewrite $1000$ as a power of $10.$ Since $1000 = 10^3,$ $\frac{1}{1000} = \frac{1}{10^3} = 10^{\text{-} 3}.$ Thus, $\log \left(\frac{1}{1000} \right)=\text{-} 3.$

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