Logarithmic Product Rule
This page follows the JoeCMath video Logarithmic Product Rule in 3 Mins. The video shows how the product rule for logarithms works in both directions: expanding one logarithm into a sum, and combining a sum of logarithms into one logarithm.
Product Rule For Logarithms
The video starts with the product rule:
\[\log_a(xy)=\log_a(x)+\log_a(y).\]The rule is used when the argument of one logarithm is a product. The conditions shown with the rule are
\[x>0,\qquad y>0,\qquad a>0,\qquad a\ne 1.\]The video treats the product rule as a two-way move: expand one logarithm when the argument is a product, or combine same-base logarithms by multiplying their arguments.
Watch this section: product rule for logarithms at 0:00.
Expanding One Logarithm
For expanding, the direction is
\[\log_a(xy)\to \log_a(x)+\log_a(y).\]The first worked example is
\[\log_{10}(5(x+1)).\]The argument is a product: $5$ multiplied by $(x+1)$. So the product rule gives
\[\log_{10}(5(x+1))=\log_{10}(5)+\log_{10}(x+1).\]The first worked expansion keeps the expression inside the parentheses together because the product rule only splits multiplication. The $x+1$ is a sum, not a product, so it stays inside one logarithm.
Watch this section: expanding logarithms using the product rule at 0:28.
More Expansion Examples
The video then shows four expansion examples together. Each one uses the same idea: find the multiplication inside the logarithm and split that product into a sum of logs.
\[\ln(x^2(x+3))=\ln(x^2)+\ln(x+3)\] \[\log_2(80\cdot 901)=\log_2(80)+\log_2(901)\] \[\log_5(x\cdot y^9)=\log_5(x)+\log_5(y^9)\] \[\log_{10}(5(x+1))=\log_{10}(5)+\log_{10}(x+1)\]The rule is not changing the base. It is only separating a product in the argument into two logarithms with the same base.
Watch this section: four expansion examples at 1:08.
Combining Logarithms
For combining, the video reverses the direction:
\[\log_a(x)+\log_a(y)\to \log_a(xy).\]The worked combining example is
\[\ln(x)+\ln(x^2-3).\]Because both terms are natural logarithms, they have the same base. The video notes that the base of natural log is $e$, even though the base is not written.
Combine by multiplying the arguments:
\[\ln(x)+\ln(x^2-3)=\ln(x(x^2-3)).\]Watch this section: combining logarithms using the product rule at 1:22.
More Combining Examples
The video then shows four combining examples together. In every row, the bases match, so the arguments can be multiplied inside one logarithm.
\[\log_{22}(40)+\log_{22}(101)=\log_{22}(40\cdot 101)\] \[\log_3(11)+\log_3(x^{10}+1)=\log_3(11(x^{10}+1))\] \[\log_5(x)+\log_5(y)=\log_5(xy)\] \[\ln(x)+\ln(x^2-3)=\ln(x(x^2-3))\]This is the same product rule as before, just read from right to left.
Watch this section: four combining examples at 1:57.
Mistakes From The Video
The last part of the video focuses on two mistakes to avoid.
First, the bases must match. For example,
\[\log_5(x)+\log_{10}(y)\]cannot be combined with the product rule because the bases are different.
Second, the product rule is about multiplication inside a logarithm, not addition. The video shows that
\[\log_{10}(x+y)\ne \log_{10}(x)+\log_{10}(y).\]The common-mistake section is careful about two boundaries: the bases must match, and the operation inside one logarithm must be multiplication.
Watch this section: common mistakes when using the product rule at 2:11.
Timestamp Guide
Related Logarithm Work
Use the logarithmic product rule topic page for more product rule notes. The logarithmic product rule example archive collects the timestamped examples from this video and related JoeCMath logarithm practice.
For the nearby logarithm rules, use the logarithms topic page, the logarithmic power rule archive, or the logarithmic quotient rule archive.
You can also watch the JoeCMath logarithms playlist: open the playlist on YouTube.