Can You Expand These Logs?
Simple overview of logarithmic expansion problems that use a combination of logarithms rules to expand the log.
Logarithmic Power Rule Review
The rule
\[\log_a(x^n)=n\log_a(x)\]Consider
\[\log_5(z^7)\]Using the logarithmic power rule we can move the power of $z$ in front of the logarithm
\[\log_5(z^7)=7\log_5(z).\]Watch this section: power rule review at 0:25.
Logarithmic Product Rule Review
The rule
\[\log_a(xy)=\log_a(x)+\log_a(y)\]Consider
\[\log_{10}(x(y+1))\]The logarithmic product rules allows us to rewrite the product of two terms inside a logarithm as the sum of two separate logarithms of the same base containing the terms from the product.
\[\log_{10}(x(y+1))=\log_{10}(x)+\log_{10}(y+1)\]Watch this section: product rule review at 0:52.
Logarithmic Quotient Rule Review
The rule
\[\log_a\left(\frac{x}{y}\right)=\log_a(x)-\log_a(y)\]Consider
\[\ln\left(\frac{z}{100}\right)\]Similar to the product rule, except the term in the denominator inside the logarithm is subtracted from the logarithm that contains the numerator.
\[\ln\left(\frac{z}{100}\right)=\ln(z)-\ln(100)\]Watch this section: quotient rule review at 1:33.
Logarithmic Product And Power Together
Consider
\[\log_2(xz^5)\]Use the product rule to separate the multiplication
\[\log_2(xz^5)=\log_2(x)+\log_2(z^5)\]Then use the power rule on the second term
\[\log_2(xz^5)=\log_2(x)+5\log_2(z)\]Watch this section: product plus power example at 2:19.
Logarithmic Quotient And Power Together
Consider
\[\log_6\left(\frac{(x+1)^4}{z^3}\right)\]Start with the quotient rule
\[\log_6\left(\frac{(x+1)^4}{z^3}\right) = \log_6((x+1)^4)-\log_6(z^3)\]Then move the powers out front using the logarithmic power rule
\[\log_6\left(\frac{(x+1)^4}{z^3}\right) = 4\log_6(x+1)-3\log_6(z)\]Watch this section: quotient plus power example at 3:53.
Logarithmic Quotient and Power Rule
Consider
\[\ln\left(\left(\frac{ma}{th}\right)^2\right)\]Since everything within the logarithm is raised to the second power we use the power rule first to move the 2 out front.
\[\ln\left(\left(\frac{ma}{th}\right)^2\right) = 2\ln\left(\frac{ma}{th}\right)\]Now apply the quotient rule
\[2\ln\left(\frac{ma}{th}\right) = 2\left(\ln(ma)-\ln(th)\right)\]Then expand each product
\[2\left(\ln(ma)-\ln(th)\right) = 2\left(\ln(m)+\ln(a)-\ln(t)-\ln(h)\right)\]So the fully expanded form is
\[2\ln(m)+2\ln(a)-2\ln(t)-2\ln(h)\]Watch this section: power, quotient, and product rules at 4:55.
A Longer Mixed Expansion
Consider
\[\log_3\left(\frac{3^3(y+3)^2}{z^3x}\right)\]Apply the quotient rule
\[\log_3(3^3(y+3)^2) - \log_3 (z^3x)\]Then apply the product rule in each term
\[\log_3(3^3)+\log_3((y+3)^2)-\log_3(z^3)-\log_3(x)\]Now use the power rule to bring all the power out front.
\[3\log_3(3)+2\log_3(y+3)-3\log_3(z)-\log_3(x)\]Since $\log_3(3)=1$, the first term becomes $3$
\[3+2\log_3(y+3)-3\log_3(z)-\log_3(x)\]Watch this section: quotient, product, inverse, and power rules at 6:47.
The Final Example (Contains Radicals)
Consider
\[\log_5\left(\frac{5\sqrt{x-2}}{(z^2+1)^3\sqrt[3]{y^5}}\right)\]The radical-heavy example is the fullest expression in the video because it asks you to translate roots, powers, products, and quotients all at once.
First rewrite the roots as powers
\[\sqrt{x-2}=(x-2)^{1/2}, \qquad \sqrt[3]{y^5}=y^{5/3}.\]Now expand the quotient and product structure
\[\log_5(5)+\log_5((x-2)^{1/2})-\log_5((z^2+1)^3)-\log_5(y^{5/3})\]Use the power rule
\[\log_5(5)+\frac{1}{2}\log_5(x-2) -3\log_5(z^2+1)-\frac{5}{3}\log_5(y)\]Finally, $\log_5(5)=1$, so the expanded form is
\[1+\frac{1}{2}\log_5(x-2) -3\log_5(z^2+1)-\frac{5}{3}\log_5(y)\]Watch this section: radicals, identity, product, quotient, and power rules at 8:55.
Timestamp Guide
Related Logarithm Work
For the rules used throughout the video, review the logarithmic product rule companion, the logarithmic quotient rule companion, and the logarithmic power rule archive.
The example archives also collect the timestamped practice from this video: product rule examples, quotient rule examples, and power rule examples.
You can also use the logarithms topic page or watch the JoeCMath logarithms playlist: open the playlist on YouTube.