Logarithmic Product Rule
Simple overview of the logarithmic product rule.
Product Rule For Logarithms
The logarithmic product rule is
\[\log_a(xy)=\log_a(x)+\log_a(y)\]When you have two factors multiplied together within your logarithm, you can rewrite it as the sum of two separate logarithms of the same base contains each of the factors.
\[\text{Assuming: }x>0,\qquad y>0,\qquad a>0,\qquad a\ne 1.\]Watch this section: product rule for logarithms at 0:00.
Expanding One Logarithm ($\rightarrow$ direction)
Let’s move from the left side to the right side of the logarithmic product rule
\[\log_a(xy)\to \log_a(x)+\log_a(y)\]Consider
\[\log_{10}(5(x+1))\]We have two factors, $5$ and $(x+1)$ multiplied together within our logarithm and so we can rewrite it as
\[\log_{10}(5(x+1))=\log_{10}(5)+\log_{10}(x+1).\]Watch this section: expanding logarithms using the product rule at 0:28.
More Expansion Examples
The following all move in the same direction, starting with a single logarithm and rewriting it as the sum of two separate logarithms.
\[\ln(x^2(x+3))\to \ln(x^2)+\ln(x+3)\] \[\log_2(80\cdot 901) \to \log_2(80)+\log_2(901)\] \[\log_5(x\cdot y^9)\to \log_5(x)+\log_5(y^9)\] \[\log_{10}(5(x+1))\to \log_{10}(5)+\log_{10}(x+1)\]Watch this section: four expansion examples at 1:08.
Combining Logarithms ($\leftarrow$ direction)
Let’s move from the right side to the left side of the logarithmic product rule
\[\log_a(xy) \leftarrow \log_a(x)+\log_a(y)\]Consider the following expression
\[\ln(x)+\ln(x^2-3)\]Since both of our logarithms have the same base (in this case the base is $e$) we can rewrite them as single logarithm by multiplying the arguments of both logs
\[\ln(x)+\ln(x^2-3)=\ln(x(x^2-3)).\]Watch this section: combining logarithms using the product rule at 1:22.
More Combining Examples
The following all move in the same direction, starting with the sum of two logarithms that have matching bases and rewriting them as a single logarithm.
\[\log_{22}(40)+\log_{22}(101) \to \log_{22}(40\cdot 101)\] \[\log_3(11)+\log_3(x^{10}+1)\to \log_3(11(x^{10}+1))\] \[\log_5(x)+\log_5(y)\to \log_5(xy)\] \[\ln(x)+\ln(x^2-3)\to \ln(x(x^2-3))\]Watch this section: four combining examples at 1:57.
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.