Certainly! Let's solve the equation step-by-step:
Given the equation:
[tex]\[ 3^m + 3^0 = 3^4 \][/tex]
### Step 1: Simplify the expression involving exponents
First, recognize that [tex]\(3^0\)[/tex] is equal to 1, since any number raised to the power of 0 is 1:
[tex]\[ 3^m + 1 = 3^4 \][/tex]
### Step 2: Simplify the right-hand side
Next, calculate [tex]\(3^4\)[/tex]. This is straightforward:
[tex]\[ 3^4 = 3 \times 3 \times 3 \times 3 = 81 \][/tex]
Therefore, our equation now looks like this:
[tex]\[ 3^m + 1 = 81 \][/tex]
### Step 3: Isolate [tex]\(3^m\)[/tex]
To isolate [tex]\(3^m\)[/tex], subtract 1 from both sides of the equation:
[tex]\[ 3^m + 1 - 1 = 81 - 1 \][/tex]
[tex]\[ 3^m = 80 \][/tex]
### Step 4: Solve for [tex]\(m\)[/tex] using logarithms
To solve for [tex]\(m\)[/tex], take the logarithm of both sides. Using the natural logarithm [tex]\( \ln \)[/tex] (you can also use common logarithm [tex]\( \log \)[/tex]), we have:
[tex]\[ \ln(3^m) = \ln(80) \][/tex]
By properties of logarithms, we can move the exponent [tex]\(m\)[/tex] to the front:
[tex]\[ m \cdot \ln(3) = \ln(80) \][/tex]
### Step 5: Isolate [tex]\(m\)[/tex]
To solve for [tex]\(m\)[/tex], divide both sides of the equation by [tex]\(\ln(3)\)[/tex]:
[tex]\[ m = \frac{\ln(80)}{\ln(3)} \][/tex]
Thus, the solution to the equation [tex]\(3^m + 1 = 81\)[/tex] is:
[tex]\[ m = \frac{\ln(80)}{\ln(3)} \][/tex]
Alternatively, in logarithmic notation, this can also be written as:
[tex]\[ m = \log_3(80) \][/tex]
So, [tex]\( m = \frac{\log(80)}{\log(3)} \)[/tex] is the detailed step-by-step solution to the given equation.
