Question

1) Log5 1/7-log√5 25/√7 2) 8 в степени log4 3-1

Answer 1

$1)\; \; log_5\, \frac{1}{7}-log_{\sqrt5}\, \frac{25}{\sqrt7}=log_5\, 7^{-1}-(log_{5^{1/2}}\, 5^2-log_{5^{1/2}}7^{1/2})=\\\\\\\star \; \; log_{a^{k}}\, b^{n}=\frac{n}{k}\, log_{a}\, b\; \; \star \\\\\\=-log_57-\frac{2}{1/2}\, log_5\, 5+log_57=-log_57-4\underbrace {log_55}_{1}+log_57=-4$

$2)\; \; 8^{log_43-1}=(2^3)^{log_{2^2}3-1}=2^{3\cdot \frac{1}{2}\cdot log_2\, 3-3}=2^{log_2\, 3^{3/2}}\cdot 2^{-3}=\\\\=3^{\frac{3}{2}}\cdot \frac{1}{2^3}=\sqrt{3^3}\cdot \frac{1}{8}=\frac{3\sqrt3}{8}$

Answer 2

I hope this helps you

Answer 3

To solve these logarithmic expressions, let's simplify each one step by step:

1. Simplify the expression: log5(1/7) - log√5(25/√7)

Step 1: Use the logarithmic identity log(a) - log(b) = log(a/b) log5(1/7) - log√5(25/√7) = log5(1/7) - (1/2) * log5(25/√7)

Step 2: Simplify the logarithms: log5(1/7) can be written as log5(7^(-1)) = -1 * log5(7) log5(25/√7) can be written as log5(25) - log5(√7) = 2 - (1/2) * log5(7)

So the expression becomes: -1 * log5(7) - (1/2) * (2 - (1/2) * log5(7))

Step 3: Continue simplifying: -1 * log5(7) - 1 + (1/4) * log5(7)

Step 4: Combine like terms: (-1 + 1/4) * log5(7) - 1 = (-3/4) * log5(7) - 1

Therefore, the simplified expression is: (-3/4) * log5(7) - 1

2. Simplify the expression: 8^(log4(3) - 1)

Step 1: Evaluate the expression inside the parentheses: log4(3) is the exponent to which 4 must be raised to get 3. In other words, 4^(log4(3)) = 3.

Step 2: Rewrite the expression with the simplified value: 8^(log4(3) - 1) = 8^(3 - 1) = 8^2 = 64

Therefore, the simplified value of the expression is 64.