Learning objective
In the same way that there are properties of exponents which allow us to simplify exponential expressions, there are properties of logarithms that allow us to simplify logarithmic expressions. In fact, each logarithm property is a consequence of an exponent property.
First, using the fact that \log_b (x)=n \iff x=b^n, it follows that: \begin{aligned}\log_b\left(b^x\right)=x&\text{ or }\ln e^x=x\\b^{\log_{b}x}=x &\text{ or }e^{\ln x}=x\end{aligned} Notice that the base of the exponent and the base of the logarithm are the same. This is called the inverse property of logarithms. We can also substitute x=0 and x=1 to get two special cases: \log_b\left(1\right)=0 \text{ or }\ln 1=0\\\log_b\left(b\right)=1\text{ or }\ln e=1
Since exponential and logarithmic equations are inverses of one another, we can derive logarithmic properties from the exponential properties.
Exponent property | Logarithm property | |
---|---|---|
Product property | b^{x}b^y=b^{x+y} | \log_b\left(xy\right) = \log_b\left(x\right)+\log_b\left(y\right) |
Quotient property | \frac{b^x}{b^y}=b^{x-y} | \log_b\left(\frac{x}{y}\right) = \log_b\left(x\right)- \log_b\left(y\right) |
Power property | \left(b^x\right)^y=b^{x\cdot y} | \log_b\left(x^p\right) = p\cdot \log_b\left(x\right) |
Equality property | b^x=b^y \iff x=y | \log_b\left(x\right)=\log_b\left(y\right)\iff x=y |
All properties associated with common logarithms can be applied to natural logarithms.
An additional property that helps simplify logarithmic expressions is the change of base property: \log_x\left(y\right) = \dfrac{\log_b\left(y\right)}{\log_b\left(x\right)}
The change of base property is best when we need to find a decimal approximation of a logarithm that cannot be simplified with the given properties. Most scientific calculators can only evaluate logarithms of base 10. We use this property to change the logarithm to base 10, then evaluate with a calculator.
Rewrite the following logarithms without products, quotients, exponents, or radicals.
\log_3 \left(5\sqrt{x}\right)
\ln \left(\dfrac{4a}{9}\right)
Rewrite the following expressions as a single logarithm.
3\log_3\left(x\right)-\log_3\left(4\right)-\log_3\left(x\right)
\ln x-5\ln y+\frac{1}{2}\ln z
Evaluate the following expressions.
\log_{8}\left(16\right)-\log_{8}\left(2\right)
\log_{2}\left(3+\sqrt{5}\right)+\log_{2}\left(3-\sqrt{5}\right)
\log_{4}8
Properties of logarithms are derived from properties of exponents. They are as follows:
Common log | Natural log | |
---|---|---|
Inverse property | \log_b\left(b^x\right) = x, b^{\log_b x}=x | \ln \left(e^x\right)= x, e^{\ln x}=x |
Product property | \log_b\left(xy\right) = \log_b\left(x\right)+\log_b\left(y\right) | \ln \left(xy\right)= \ln\left(x\right)+\ln\left(y\right) |
Quotient property | \log_b\left(\frac{x}{y}\right) = \log_b\left(x\right)- \log_b\left(y\right) | \ln \left(\frac{x}{y}\right)= \ln\left(x\right)-\ln\left(y\right) |
Power property | \log_b\left(x^p\right) = p\log_b\left(x\right) | \ln\left(x^p\right) = p\ln\left(x\right) |
Equality property | \log_b\left(x\right)=\log_b\left(y\right)\iff \\x=y | \ln\left(x\right)=\ln\left(y\right)\iff \\x=y |
Change of base property | \log_b\left(x\right) = \dfrac{\log_a\left(x\right)}{\log_a\left(b\right)} |
The properties of logarithms have direct connections to transformations in the graph of a logarithmic function.
The product property, which states that log_b(xy) = log_b(x) + log_b(y), implies that a horizontal dilation of a logarithmic function, f(x) = log_b(kx), equates to a vertical translation. Essentially, this means multiplying the input by a constant (k) results in the graph shifting up or down by a distance related to the logarithm of that constant.
The power property, which states that log_{b} x^n= n \cdot log_bx , tells us that raising the input of a logarithmic function to a power, f(x) = log_bx^k, results in a vertical dilation by a factor of k.
The change of base property, which states that log_{b} x= \dfrac{\log_{a}x}{log_{a}b}, demonstrates that all logarithmic functions are vertical dilations of each other.
Given the logarithmic function f\left(x\right) = \log_{2}x, identify the vertical translation that is equivalent to the horizontal dilation f\left(x\right) = \log_{2}3x.
The properties of logarithms can be visually represented as transformations on the graph of a logarithmic function.