Can You Expand These Logs?
This page follows the JoeCMath video Can You Expand These Logs? The video works like a guided mixed-practice set: start with one logarithm rule, then add another rule only when the expression actually needs it.
Power Rule Review
The video begins with the logarithmic power rule:
\[\log_a(x^n)=n\log_a(x).\]The first review example is
\[\log_5(z^7).\]The exponent on $z$ moves in front of the logarithm:
\[\log_5(z^7)=7\log_5(z).\]That is the basic move the rest of the video keeps reusing whenever a power appears inside a logarithm.
Watch this section: power rule review at 0:25.
Product Rule Review
The next review is the product rule:
\[\log_a(xy)=\log_a(x)+\log_a(y).\]The example in the video is
\[\log_{10}(x(y+1)).\]The argument is a product: $x$ multiplied by $(y+1)$. The product rule gives
\[\log_{10}(x(y+1))=\log_{10}(x)+\log_{10}(y+1).\]The expression $y+1$ stays together because the rule splits multiplication, not addition.
Watch this section: product rule review at 0:52.
Quotient Rule Review
The quotient rule turns division inside one logarithm into subtraction:
\[\log_a\left(\frac{x}{y}\right)=\log_a(x)-\log_a(y).\]The review example is
\[\ln\left(\frac{z}{100}\right).\]The numerator becomes the positive logarithm and the denominator becomes the subtracted logarithm:
\[\ln\left(\frac{z}{100}\right)=\ln(z)-\ln(100).\]This sets up the later examples where several denominator factors all become subtraction terms.
Watch this section: quotient rule review at 1:33.
Product And Power Together
The first mixed-rule example is
\[\log_2(xz^5).\]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:
\[\log_2(xz^5)=\log_2(x)+5\log_2(z).\]The video is building the expansion one visible rule at a time: product first, then power.
Watch this section: product plus power example at 2:19.
Quotient And Power Together
The next example is
\[\log_6\left(\frac{(x+1)^4}{z^3}\right).\]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:
\[\log_6\left(\frac{(x+1)^4}{z^3}\right) =4\log_6(x+1)-3\log_6(z).\]The subtraction comes from the denominator. The coefficients come from the exponents.
Watch this section: quotient plus power example at 3:53.
A Quotient Raised To A Power
The video then gives an example where the entire quotient is squared:
\[\ln\left(\left(\frac{ma}{th}\right)^2\right).\]The point is to move the outside exponent first, then let the product and quotient rules separate the pieces:
\[\ln\left(\left(\frac{ma}{th}\right)^2\right) =2\ln\left(\frac{ma}{th}\right).\]Now expand the quotient:
\[2\ln\left(\frac{ma}{th}\right) =2\left(\ln(ma)-\ln(th)\right).\]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
The next expression is
\[\log_3\left(\frac{3^3(y+3)^2}{z^3x}\right).\]The numerator has a product, and the denominator has a product. Expanding the quotient gives
\[\log_3(3^3)+\log_3((y+3)^2)-\log_3(z^3)-\log_3(x).\]Now use the power rule:
\[3\log_3(3)+2\log_3(y+3)-3\log_3(z)-\log_3(x).\]Since $\log_3(3)=1$, the first term becomes $3$:
\[3+2\log_3(y+3)-3\log_3(z)-\log_3(x).\]This is the first place in the video where a logarithm identity simplifies one term after the expansion.
Watch this section: quotient, product, inverse, and power rules at 6:47.
The Radical Example
The final worked expression is
\[\log_5\left(\frac{5\sqrt{x-2}}{(z^2+1)^3\sqrt[3]{y^5}}\right).\]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.