We are given four equations and asked to solve for $x$ in each equation and round the answer to 3 decimal places. The four equations are: 1. $11^{10x-15} = 12$

1. Problem Description

We are given four equations and asked to solve for $x$ in each equation and round the answer to 3 decimal places.

The four equations are:

1. $11^{10x-15} = 12$

2. $\log_4(x^6) = 5$

3. $\frac{10^{14x}}{10^{11x-5}} = 9$

4. $\log_2(3x-7) + \log_2(4) = 5$

2. Solution Steps

Equation 1: $11^{10x-15} = 12$

Take the natural logarithm of both sides:

$\ln(11^{10x-15}) = \ln(12)$

$(10x-15)\ln(11) = \ln(12)$

$10x-15 = \frac{\ln(12)}{\ln(11)}$

$10x = 15 + \frac{\ln(12)}{\ln(11)}$

$x = \frac{1}{10}\left(15 + \frac{\ln(12)}{\ln(11)}\right)$

$x = \frac{1}{10}\left(15 + \frac{2.4849}{2.3979}\right)$

$x = \frac{1}{10}(15 + 1.0363)$

$x = \frac{1}{10}(16.0363)$

$x = 1.60363$

Rounded to 3 decimal places: $x = 1.604$

Equation 2: $\log_4(x^6) = 5$

$x^6 = 4^5$

$x^6 = 1024$

$x = (1024)^{\frac{1}{6}}$

$x = (2^{10})^{\frac{1}{6}}$

$x = 2^{\frac{10}{6}} = 2^{\frac{5}{3}}$

$x = 2^{1.6666...}$

$x = 3.1748$

Rounded to 3 decimal places: $x = 3.175$

Equation 3: $\frac{10^{14x}}{10^{11x-5}} = 9$

$10^{14x - (11x-5)} = 9$

$10^{14x - 11x + 5} = 9$

$10^{3x+5} = 9$

Take the logarithm base 10 of both sides:

$\log_{10}(10^{3x+5}) = \log_{10}(9)$

$3x+5 = \log_{10}(9)$

$3x = \log_{10}(9) - 5$

$x = \frac{\log_{10}(9) - 5}{3}$

$x = \frac{0.9542 - 5}{3}$

$x = \frac{-4.0458}{3}$

$x = -1.3486$

Rounded to 3 decimal places: $x = -1.349$

Equation 4: $\log_2(3x-7) + \log_2(4) = 5$

$\log_2((3x-7)(4)) = 5$

$\log_2(12x-28) = 5$

$12x-28 = 2^5$

$12x-28 = 32$

$12x = 32+28$

$12x = 60$

$x = \frac{60}{12}$

$x = 5$

3. Final Answer

1. $x = 1.604$

2. $x = 3.175$

3. $x = -1.349$

4. $x = 5.000$