Question

rewrite the logarithmic expression as a single logarithm with the same base. assume all expressions exist and are well - defined. simplify any fractions. $8\log_{t}2 - \log_{t}10$

Explanation:

Step1: Apply Power Rule

The power rule of logarithms states that \( n\log_b a=\log_b a^n \). For \( 8\log_t 2 \), we apply this rule:
\( 8\log_t 2=\log_t 2^8 \)
Calculate \( 2^8 = 256 \), so \( 8\log_t 2=\log_t 256 \)

Step2: Apply Quotient Rule

The quotient rule of logarithms states that \( \log_b a-\log_b c = \log_b\frac{a}{c} \). Now we have \( \log_t 256-\log_t 10 \), applying the quotient rule:
\( \log_t 256-\log_t 10=\log_t\frac{256}{10} \)

Step3: Simplify the Fraction

Simplify \( \frac{256}{10} \) by dividing numerator and denominator by 2:
\( \frac{256\div2}{10\div2}=\frac{128}{5} \)
So the expression becomes \( \log_t\frac{128}{5} \)

Answer:

\(\log_{t}\frac{128}{5}\)