Manipulating Logarithmic Functions (College Board AP® Precalculus): Revision Note

Properties of logarithms

What is the product property for logarithms?

• The product property for logarithms states that

  • ${\log }_b\left(xy\right)={\log }_{b}x+{\log }_{b}y$

    • The logarithm of a product equals the sum of the logarithms

  • This works in reverse too: a sum of logarithms (with the same base) can be combined into a single logarithm of a product

    • E.g. ${\log }_{3}4+{\log }_{3}7={\log }_3(4·7)={\log }_{3}28$

• There is also a corresponding quotient property

  • ${\log }_b\left(\frac{x}{y}\right)={\log }_{b}x-{\log }_{b}y$

    • The logarithm of a quotient equals the difference of the logarithms

  • This also works in reverse, to combine a difference of logarithms

    • E.g. $\ln 15-\ln 3=\ln \frac{15}{3}=\ln 5$

• The product property has an important graphical implication

  • Every horizontal dilation of a logarithmic function, $f\left(x\right)={\log }_b\left(kx\right)$

  • is equivalent to a vertical translation

  • because ${\log }_b\left(kx\right)={\log }_{b}k+{\log }_{b}x$

    • The constant ${\log }_{b}k$ acts as a vertical shift of ${\log }_{b}x$

What is the power property for logarithms?

• The power property for logarithms states that

  • ${\log }_b\left(x^n\right)=n{\log }_{b}x$

    • An exponent inside a logarithm can be brought out as a multiplier in front

    • E.g. ${\log }_5\left(x^2\right)=2{\log }_{5}x$

  • This works in reverse too: a coefficient in front of a logarithm can be moved inside as an exponent

    • E.g. $3{\log }_{2}x={\log }_2\left(x^3\right)$

    • E.g. $\frac{1}{2}\ln x=\ln \left(x^{1/2}\right)=\ln \sqrt{x}$

• This also has an important graphical implication

  • Raising the input of a logarithmic function to a power, $f\left(x\right)={\log }_b\left(x^k\right)$

  • is equivalent to a vertical dilation by a factor of $k$

  • because ${\log }_b\left(x^k\right)=k{\log }_{b}x$

What is the change of base property?

• The change of base property for logarithms states that

  • ${\log }_{b}x=\frac{{\log }_{a}x}{{\log }_{a}b}$

    • where $a>0$ and $a\ne 1$

• This allows you to convert a logarithm from one base to another

  • E.g. ${\log }_{5}12=\frac{\ln 12}{\ln 5}=\frac{\log 12}{\log 5}$

  • You can use any base for the conversion

    • $e$ and 10 are both commonly-used 'standard' bases

• An important graphical implication of this is that all logarithmic functions are vertical dilations of each other

  • Since ${\log }_{b}x=\frac{1}{{\log }_{a}b}·{\log }_{a}x$

    • the function ${\log }_{b}x$ is just a constant multiple of ${\log }_{a}x$

  • This means changing the base of a logarithmic function only stretches or compresses the graph vertically

    • It doesn't change the overall shape

How do you use these properties to rewrite expressions?

• On the exam, you are frequently asked to combine multiple logarithmic terms into a single logarithm

  • or to expand a single logarithm into multiple terms

• For combining (multiple terms → single logarithm):

  • First use the power property to move any coefficients inside as exponents

    • E.g. $2{\log }_{3}x+{\log }_{3}5-3{\log }_{3}y={\log }_3\left(x^2\right)+{\log }_{3}5-{\log }_3(y^3)$

  • Then use the product property (for addition) and quotient property (for subtraction) to combine into one logarithm

    • $={\log }_3\left(5x^2\right)-{\log }_3(y^3)$

    • $={\log }_3\left(\frac{5x^2}{y^3}\right)$

• When combining terms, make sure all logarithms have the same base before applying the product or quotient properties

  • If the bases differ, use the change of base property first

• For expanding (single logarithm → multiple terms):

  • Apply the product/quotient/power properties in reverse

  • E.g. ${\log }_2\left(\frac{x^3\sqrt{y}}{z}\right)={\log }_2\left(x^3\right)+{\log }_2\left(\sqrt{y}\right)-{\log }_{2}z=3{\log }_{2}x+\frac{1}{2}{\log }_{2}y-{\log }_{2}z$