Sure, let's convert each logarithmic equation into its equivalent exponential form step by step:
### 1. [tex]\(\log_2 32 = 5\)[/tex]
The logarithmic equation [tex]\(\log_2 32 = 5\)[/tex] means that 2 raised to the power of 5 equals 32. In exponential form, it is written as:
[tex]\[ 2^5 = 32 \][/tex]
### 2. [tex]\(\log_3 81 = 4\)[/tex]
The logarithmic equation [tex]\(\log_3 81 = 4\)[/tex] means that 3 raised to the power of 4 equals 81. In exponential form, it is written as:
[tex]\[ 3^4 = 81 \][/tex]
### 3. [tex]\(\ln \frac{1}{e} = -1\)[/tex]
The natural logarithm [tex]\(\ln \frac{1}{e} = -1\)[/tex] means that [tex]\(e\)[/tex] (Euler's number, approximately 2.718) raised to the power of -1 equals [tex]\(\frac{1}{e}\)[/tex]. In exponential form, it is written as:
[tex]\[ e^{-1} = \frac{1}{e} \][/tex]
### 4. [tex]\(\log_2 \frac{1}{8} = -3\)[/tex]
The logarithmic equation [tex]\(\log_2 \frac{1}{8} = -3\)[/tex] means that 2 raised to the power of -3 equals [tex]\(\frac{1}{8}\)[/tex]. In exponential form, it is written as:
[tex]\[ 2^{-3} = \frac{1}{8} \][/tex]
### 5. [tex]\(\log 100,000 = 5\)[/tex]
The common logarithm [tex]\(\log 100,000 = 5\)[/tex] means that 10 raised to the power of 5 equals 100,000. In exponential form, it is written as:
[tex]\[ 10^5 = 100,000 \][/tex]
### 6. [tex]\(\log 0.001 = -3\)[/tex]
The common logarithm [tex]\(\log 0.001 = -3\)[/tex] means that 10 raised to the power of -3 equals 0.001. In exponential form, it is written as:
[tex]\[ 10^{-3} = 0.001 \][/tex]
