Logarithms
A logarithm is the inverse function to an exponential function. For example, for the exponential function [latex]y = 2^x[/latex], if we have an input of [latex]x = 6[/latex], we get an output of [latex]y = 64[/latex], and we write [latex]64 = 2^6[/latex]. The logarithmic function [latex]y = \log_2(x)[/latex] is the reverse of this. We swap the input and the output, so now [latex]x = 64[/latex] and [latex]y = 6[/latex]. We see [latex]6 = \log_2(64)[/latex].
In calculus, we will mostly use the exponential function [latex]e^x[/latex] and its inverse, [latex]\ln(x)[/latex]. Below are some important formulas:
\begin{align*}
e^{\ln(x)} & = x \\
\ln(e^x) & = x \\
\ln(x) + \ln(y) & = \ln(xy) \\
\ln(x) – \ln(y) & = \ln\left(\frac{x}{y}\right) \\
a \ln(x) & = \ln(x^a)
\end{align*}
Examples:
There are two ways to do this one. First, we can bring down the exponent of two down in front [latex]\ln(x^2) = 2 \ln(x)[/latex]. Then can combine the like terms of [latex]2\ln(x)[/latex] and [latex]\ln(x)[/latex]:
\begin{align*}
\ln(x^2) – \ln(x) & = 2 \ln(x) – \ln(x) \\
& = \boxed{\ln(x)}
\end{align*}
Alternatively, we can rewrite the subtraction as a division, like so:
\begin{align*}
\ln(x^2) – \ln(x) & = \ln\left(\frac{x^2}{x}\right) \\
& = \boxed{\ln(x)}
\end{align*}
Either way we get the same answer!
First, we rewrite the multiplication using addition. Then we can simply from there.
\begin{align*}
\ln(e^3 x^4) – 3 \ln(x) & = \ln(e^3) + \ln(x^4) – 3 \ln(x)\\
& = 3 + 4 \ln(x) – 3 \ln(x) \\
& = \boxed{3 + \ln(x)}
\end{align*}
We know [latex]\sqrt{x} = x^{1/2}[/latex], so [latex]\ln\left(\sqrt{x}\right) = \ln\left(x^{1/2}\right) = \boxed{\frac{1}{2} \ln(x)}[/latex].
We can rewrite all the products and divisions as addition and subtraction:
\begin{align*}
\ln\left(\frac{\sqrt{x} y}{z^3}\right) – \ln\left(\frac{z}{\sqrt{x} y^3}\right) & = \ln(\sqrt{x}) + \ln(y) – \ln(z^3) – [\ln(z) – \ln(\sqrt{x}) – \ln(y^3)] \\
& = \frac{1}{2} \ln(x) + \ln(y) – 3 \ln(z) – \ln(z) + \frac{1}{2} \ln(x) + 3 \ln(y) \\
& = \boxed{\ln(x) + 4 \ln(y) – 4 \ln(z)}.
\end{align*}