We are given that: \[ \log_5 2 = x \] We need to find: \[ \log_{25} 800 \] Step 1: Express in Terms of Base 5
We rewrite the base 25 logarithm using: \[ \log_{25} a = \frac{\log_5 a}{\log_5 25} \] Since \( 25 = 5^2 \), we know: \[ \log_5 25 = 2 \] Thus, \[ \log_{25} 800 = \frac{\log_5 800}{2} \] Step 2: Express \( \log_5 800 \)
Factorize 800: \[ 800 = 2^5 \times 5^2 \] Taking log base 5: \[ \log_5 800 = \log_5 (2^5 \times 5^2) \] Using logarithm properties: \[ \log_5 800 = 5 \log_5 2 + \log_5 5^2 \] \[ = 5x + 2 \] Step 3: Compute \( \log_{25} 800 \) \[ \log_{25} 800 = \frac{5x + 2}{2} \] \[ = 2.5x + 1 \]