Question

use the properties of logarithms to evaluate each of the following expressions. (a) $log_{5} 45 - 2log_{5} 3 = square$ (b) $ln e^{-9} + ln e^{5} = square$

Explanation:

Part (a)

Step1: Bring coefficient as exponent

$2\log_{5}3 = \log_{5}3^2 = \log_{5}9$

Step2: Apply log subtraction rule

$\log_{5}45 - \log_{5}9 = \log_{5}\frac{45}{9}$

Step3: Simplify the argument

$\log_{5}\frac{45}{9} = \log_{5}5$

Step4: Evaluate the logarithm

$\log_{5}5 = 1$

Part (b)

Step1: Bring exponents as coefficients

$\ln e^{-9} = -9\ln e$, $\ln e^{5}=5\ln e$

Step2: Evaluate $\ln e$ (equals 1)

$-9(1) + 5(1) = -9 + 5$

Step3: Compute the sum

$-9 + 5 = -4$

Answer:

(a) $1$
(b) $-4$