Expand the logarithmic expression log((a^2)/(10*b))

 Exercise

$\log\left(\frac{a^2}{10b}\right)$

 

The difference of two logarithms of equal base $b$ is equal to the logarithm of the quotient: $\log_b(x)-\log_b(y)=\log_b\left(\frac{x}{y}\right)$

$\log \left(a^2\right)-\log \left(10b\right)$

 

Use the product rule for logarithms: $\log_b\left(MN\right)=\log_b\left(M\right)+\log_b\left(N\right)$, where $M=b$ and $N=10$

$\log \left(a^2\right)-\left(\log \left(b\right)+\log \left(10\right)\right)$

 

Evaluating the logarithm of base $10$ of $10$

$\log \left(a^2\right)-\left(\log \left(b\right)+1\right)$

 

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$2\log \left(a\right)-\left(\log \left(b\right)+1\right)$

 

Simplify the product $-(\log \left(b\right)+1)$

$2\log \left(a\right)-\log \left(b\right)-1$

$2\log \left(a\right)-\log \left(b\right)-1$

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