3.4: Algebra Tips and Tricks Part VI (Logarithms)
Logarithms
A logarithm is the inverse function to an exponential function. For example, for the exponential function \(y = 2^x\), if we have an input of \(x = 6\), we get an output of \(y = 64\), and we write \(64 = 2^6\). The logarithmic function \(y = \log_2(x)\) is the reverse of this. We swap the input and the output, so now \(x = 64\) and \(y = 6\). We see \(6 = \log_2(64)\).
In calculus, we will mostly use the exponential function \(e^x\) and its inverse, \(\ln(x)\). 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 \(\ln(x^2) = 2 \ln(x)\). Then can combine the like terms of \(2\ln(x)\) and \(\ln(x)\):
\[\begin{align*} \ln(x^2) - \ln(x) & = 2 \ln(x) - \ln(x) \\ & = \boxed{\ln(x)} \end{align*}\]
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
, so
.
We can rewrite all the products and divisions as addition and subtraction:
\[\begin{align*} \ln\left(\frac{\sqrt{x} y}{z^3}\right) - \ln\left(\frac{z}{\sqrt{x} y^3}\right) & = \ln(\sqrt{x}) + \ln(y) - \ln(z^3) - [\ln(z) - \ln(\sqrt{x}) - \ln(y^3)] \\ & = \frac{1}{2} \ln(x) + \ln(y) - 3 \ln(z) - \ln(z) + \frac{1}{2} \ln(x) + 3 \ln(y) \\ & = \boxed{\ln(x) + 4 \ln(y) - 4 \ln(z)}. \end{align*}\]