Skill: Apply logarithm laws

Explanation and Free Practice Resources

Logarithm laws express multiplication, division, and powers of positive quantities as addition, subtraction, and multiplication of logarithms: $\log_b(MN)=\log_b M+\log_b N$ , $\log_b(M/N)=\log_b M-\log_b N$ , and $\log_b(M^p)=p\log_b M$ , for $b>0,\ b\ne1$ and valid positive arguments. This understanding supports expanding and condensing logarithmic expressions and transforming equivalent equations, while distinguishing these laws from the invalid claim that $\log_b(M+N)$ splits into a sum; complex logarithms and more advanced generalizations are outside this scope.

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Detailed Explanation: Apply logarithm laws

Logarithm laws let you rewrite products, quotients, and powers as sums, differences, and multiples:

\log_b(MN)=\log_b M+\log_b N

\log_b\left(\frac{M}{N}\right)=\log_b M-\log_b N

\log_b(M^p)=p\log_b M

These rules require $(b>0)$ , $(b\ne 1)$ , and positive logarithm arguments.

Example: Expand

\log_3\left(\frac{x^2y^3}{z}\right), \qquad x,y,z>0

Step 1: Apply the quotient law.

\log_3\left(\frac{x^2y^3}{z}\right) = \log_3(x^2y^3)-\log_3 z

Step 2: Apply the product law to the numerator.

\log_3(x^2y^3) = \log_3(x^2)+\log_3(y^3)

So,

\log_3\left(\frac{x^2y^3}{z}\right) = \log_3(x^2)+\log_3(y^3)-\log_3 z

Step 3: Apply the power law.

\log_3(x^2)=2\log_3 x \qquad\text{and}\qquad \log_3(y^3)=3\log_3 y

Therefore,

\boxed{\log_3\left(\frac{x^2y^3}{z}\right) = 2\log_3 x+3\log_3 y-\log_3 z}

Remember that logarithm laws do not apply to sums:

\log_b(M+N)\ne \log_b M+\log_b N

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Logarithms - Quotient Property - Division as Fraction To Difference (Variables)

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