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Apply logarithm laws

Logarithm laws express multiplication, division, and powers of positive quantities as addition, subtraction, and multiplication of logarithms: log⁡b(MN)=log⁡bM+log⁡bN\log_b(MN)=\log_b M+\log_b N, log⁡b(M/N)=log⁡bM−log⁡bN\log_b(M/N)=\log_b M-\log_b N, and log⁡b(Mp)=plog⁡bM\log_b(M^p)=p\log_b M, for b>0, b≠1b>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)\log_b(M+N) splits into a sum; complex logarithms and more advanced generalizations are outside this scope.

Detailed Explanation: Apply logarithm laws

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

log⁡b(MN)=log⁡bM+log⁡bN\log_b(MN)=\log_b M+\log_b N log⁡b(MN)=log⁡bM−log⁡bN\log_b\left(\frac{M}{N}\right)=\log_b M-\log_b N log⁡b(Mp)=plog⁡bM\log_b(M^p)=p\log_b M

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

Example: Expand

log⁡3(x2y3z),x,y,z>0\log_3\left(\frac{x^2y^3}{z}\right), \qquad x,y,z>0

Step 1: Apply the quotient law.

log⁡3(x2y3z)=log⁡3(x2y3)−log⁡3z\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(x2y3)=log⁡3(x2)+log⁡3(y3)\log_3(x^2y^3) = \log_3(x^2)+\log_3(y^3)

So,

log⁡3(x2y3z)=log⁡3(x2)+log⁡3(y3)−log⁡3z\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(x2)=2log⁡3xandlog⁡3(y3)=3log⁡3y\log_3(x^2)=2\log_3 x \qquad\text{and}\qquad \log_3(y^3)=3\log_3 y

Therefore,

log⁡3(x2y3z)=2log⁡3x+3log⁡3y−log⁡3z\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)≠log⁡bM+log⁡bN\log_b(M+N)\ne \log_b M+\log_b N

Learn by doing: Apply logarithm laws

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