Rewriting expressions with power and logarithm properties

Statement

Rewrite the following expressions in an equivalent form (do not evaluate them), using the properties of powers and logarithms:

• (a) $5^{23}\cdot 5^{17}$
• (b) $\left(13^{8}\right)^{9}$
• (c) $\left(3\cdot\dfrac{4}{5}\right)^{12}$
• (d) $\log(45\cdot 54)$
• (e) $\log\left(\dfrac{45}{54}\right)$
• (f) $\log\left(14^{123}\right)$

Solution

(a) Same base, add the exponents: $5^{23}\cdot 5^{17}=5^{40}$.

(b) Power of a power, multiply the exponents: $\left(13^{8}\right)^{9}=13^{72}$.

(c) The product is raised to the same power: $\left(3\cdot\dfrac{4}{5}\right)^{12}=3^{12}\cdot\dfrac{4^{12}}{5^{12}}$.

(d) Logarithm of a product: $\log(45\cdot 54)=\log 45+\log 54$.

(e) Logarithm of a quotient: $\log\left(\dfrac{45}{54}\right)=\log 45-\log 54$.

(f) The exponent comes out of the logarithm: $\log\left(14^{123}\right)=123\log 14$.

$\boxed{\text{(a) }5^{40};\ \text{(b) }13^{72};\ \text{(c) }3^{12}\dfrac{4^{12}}{5^{12}};\ \text{(d) }\log 45+\log 54;\ \text{(e) }\log 45-\log 54;\ \text{(f) }123\log 14}$