The problem asks to express each given expression as a single logarithm. (a) $2log_a5 + log_a4 - log_a10$ (b) $2log_b3 - log_b15 + log_b5$ (c) $log_512 - (\frac{1}{2}log_59 + \frac{1}{3}log_58)$

1. Problem Description

The problem asks to express each given expression as a single logarithm.

(a) $2log_a5 + log_a4 - log_a10$

(b) $2log_b3 - log_b15 + log_b5$

(c) $log_512 - (\frac{1}{2}log_59 + \frac{1}{3}log_58)$

2. Solution Steps

(a)

First, use the power rule of logarithms: $nlog_ax = log_ax^n$.

$2log_a5 = log_a5^2 = log_a25$

Then the expression becomes: $log_a25 + log_a4 - log_a10$

Use the product rule of logarithms: $log_ax + log_ay = log_axy$.

$log_a25 + log_a4 = log_a(25 \times 4) = log_a100$

Then the expression becomes: $log_a100 - log_a10$

Use the quotient rule of logarithms: $log_ax - log_ay = log_a(\frac{x}{y})$.

$log_a100 - log_a10 = log_a(\frac{100}{10}) = log_a10$

(b)

First, use the power rule of logarithms: $nlog_ax = log_ax^n$.

$2log_b3 = log_b3^2 = log_b9$

Then the expression becomes: $log_b9 - log_b15 + log_b5$

Use the quotient rule of logarithms: $log_ax - log_ay = log_a(\frac{x}{y})$.

$log_b9 - log_b15 = log_b(\frac{9}{15}) = log_b(\frac{3}{5})$

Then the expression becomes: $log_b(\frac{3}{5}) + log_b5$

Use the product rule of logarithms: $log_ax + log_ay = log_axy$.

$log_b(\frac{3}{5}) + log_b5 = log_b(\frac{3}{5} \times 5) = log_b3$

(c)

First, use the power rule of logarithms: $nlog_ax = log_ax^n$.

$\frac{1}{2}log_59 = log_59^{\frac{1}{2}} = log_5\sqrt{9} = log_53$

$\frac{1}{3}log_58 = log_58^{\frac{1}{3}} = log_5\sqrt[3]{8} = log_52$

Then the expression becomes: $log_512 - (log_53 + log_52)$

Use the product rule of logarithms: $log_ax + log_ay = log_axy$.

$log_53 + log_52 = log_5(3 \times 2) = log_56$

Then the expression becomes: $log_512 - log_56$

Use the quotient rule of logarithms: $log_ax - log_ay = log_a(\frac{x}{y})$.

$log_512 - log_56 = log_5(\frac{12}{6}) = log_52$

3. Final Answer

(a) $log_a10$

(b) $log_b3$

(c) $log_52$