Questions: Express the given logarithm as a sum and/or difference of logarithms. Simplify, if possible. Assume that all variables represent positive real numbers. log8 (3/19) log8 19 - log8 3 log8 3 - log8 19 log8 3 ÷ log8 19 log8 3 + log8 3

Solution Steps

To express the given logarithm as a sum and/or difference of logarithms, we can use the properties of logarithms. Specifically, the quotient rule for logarithms states that \(\log_b \frac{M}{N} = \log_b M - \log_b N\). Applying this rule to the given expression \(\log_8 \frac{3}{19}\), we can rewrite it as a difference of two logarithms.

We start with the logarithmic expression: \[ \log_8 \frac{3}{19} \] Using the quotient rule for logarithms, we can express this as: \[ \log_8 \frac{3}{19} = \log_8 3 - \log_8 19 \]

Next, we calculate the individual logarithms: \[ \log_8 3 \approx 0.5283 \] \[ \log_8 19 \approx 1.4160 \]

Now, we find the difference: \[ \log_8 \frac{3}{19} = \log_8 3 - \log_8 19 \approx 0.5283 - 1.4160 \approx -0.8877 \]

Final Answer