Sure! Let's go through the steps to solve the equation [tex]\(\log_5(125x) = 4\)[/tex].
### Step 1: Understanding the Logarithmic Equation
The given equation is:
[tex]\[ \log_5(125x) = 4 \][/tex]
This can be interpreted as: "The logarithm base 5 of [tex]\(125x\)[/tex] equals 4."
### Step 2: Converting the Logarithmic Equation to Exponential Form
To solve for [tex]\(x\)[/tex], let's convert the logarithmic equation to its exponential form. Recall that if [tex]\(\log_b A = C\)[/tex], then [tex]\(b^C = A\)[/tex]. In this case, [tex]\(b = 5\)[/tex], [tex]\(A = 125x\)[/tex], and [tex]\(C = 4\)[/tex].
Thus,
[tex]\[ 5^4 = 125x \][/tex]
### Step 3: Evaluating the Exponent
Next, let's evaluate [tex]\(5^4\)[/tex]:
[tex]\[ 5^4 = 5 \times 5 \times 5 \times 5 = 625 \][/tex]
So, the equation becomes:
[tex]\[ 625 = 125x \][/tex]
### Step 4: Solving for [tex]\(x\)[/tex]
To find [tex]\(x\)[/tex], divide both sides of the equation by 125:
[tex]\[ x = \frac{625}{125} \][/tex]
### Step 5: Simplifying the Fraction
Simplify [tex]\(\frac{625}{125}\)[/tex]:
[tex]\[ \frac{625}{125} = 5 \][/tex]
Therefore, the solution to the equation [tex]\(\log_5(125x) = 4\)[/tex] is:
[tex]\[ x = 5 \][/tex]
