Precalculus Review Materials

From our previous work, we know that if $b>0$ then the range of the exponential function $f(x)=b(x)$ is the positive real numbers. If $b>0$ then the graph of $f$ increases as we move right on the $x$ axis, and if $0<b<1$ then it increases as we move left on the $x$-axis. (The function $f(x)=b^x$ is not very exciting if $b=1\text{.}$) In either case, that means for any positive real number $y$ there exists a unique real number $x$ such that $b^x=y\text{.}$ This number is the base-$b$ logarithm of $y$, and denoted by ${\log }_b(y)\text{.}$ That is, if $b>0$ and $b\ne 1\text{,}$ then ${\log }_b(y)$ is the power to which $b$ must be raised in order to get the positive number $y\text{.}$

It is important to notice that ${\log }_b(y)$ is only defined for positive numbers $y\text{.}$ Logarithms are exponents. By definition $b^{{\log }_b(y)}=y\text{.}$ Since $b$ is positive, so is any power of $b\text{.}$ Thus ${\log }_b(y)$ is undefined when $y$ is negative because no power of a positive number can give us a negative number.

Logarithms have properties that follow immediately from the fact that they are exponents.

Properties of Logarithms. Suppose $b>0$ and $b\ne 1\text{.}$

When the base $b$ is omitted, as in $\log (100)\text{,}$ it is assumed to be 10. Logarithms to base 10 are called common logarithms. Logarithms to base $e$ are called natural logarithms, and denoted by $ln(x)$ rather than $lo{g}_e(x)\text{.}$

For a positive real number $b\ne 1\text{,}$ the logarithm function with base $b$ is the function $f(x)={\log }_b(x)\text{.}$ Its domain is the set of positive real numbers. Its range is the set of all real numbers.

Notice that the functions $f(x)=b^x$ and $g(y)={\log }_b(y)$ are inverses by definition: $b^x=y$ if and only if ${\log }_b(y)=x\text{.}$ Thus, if $b\ne 1\text{,}$ the logarithm function with base $b$ is the inverse of the exponential function with base $b\text{.}$ (The function $f(x)=1^x$ does not have an inverse because, for example $1^2=1^3=1\text{.}$)

Finally, we observe that logarithm functions with different bases are just multiples of each other. We know $b^{{\log }_b(x)}=x\text{.}$ Therefore, ${\log }_a\left(b^{{\log }_b(x)}\right)={\log }_a(x)\text{.}$ From one of the properties of logarithms, we know that ${\log }_a\left(b^{{\log }_b(x)}\right)={\log }_b(x){\log }_a(b)\text{,}$ and so this is the same as ${\log }_b(x){\log }_a(b)={\log }_a(x)\text{.}$ Since ${\log }_a(b)$ is a number, this says that ${\log }_a(x)$ is a multiple of ${\log }_b(x)\text{.}$

The graphs of $f(x)=log(x)$ and $g(x)=\ln (x)$ are shown below.

The same principles as before of shifting graphs or stretching them vertically or horizontally apply.