Question

1. evaluate $0.01^{\log_{10}(2)}$
2. expand as sums and differences

$\ln\left(\frac{e^3 x^4}{e^2 x^{-8}}\
ight)$

1. solve for $x$

$3^{x + 5} = 27^{-2x + 1}$

1. solve for $x$

$\log_2(1 - x) = 2 + \log_2(1 + x)$

1. evaluate $\ln(e^4)$
2. evaluate $\sqrt{\log_{10}(100)} - \log_{10}\sqrt{100}$
3. evaluate $e^{\ln 3}$
4. evaluate $\log_{10}(\log_{10}(10))$
5. suppose $x = \log(a)$ and $y = \log(b)$. write the following expressions in terms of $x$ and $y$

a) $\log(ab)$
b) $\log(a)\log(b)$
c) $\log(a/(b^2))$

1. solve for $x$

$2^x = 32$

Explanation:

Let's solve question 17: \( 2^x = 32 \)

Step1: Express 32 as a power of 2

We know that \( 32 = 2^5 \), so the equation \( 2^x = 32 \) can be rewritten as \( 2^x = 2^5 \).

Step2: Use the property of exponential functions

For exponential functions with the same base, if \( a^m = a^n \) (where \( a > 0, a
eq 1 \)), then \( m = n \). Here, the base \( a = 2 \), so we can set the exponents equal to each other: \( x = 5 \).

Answer:

\( x = 5 \)