Questions: Write the equation 5^(3x) = 71 in logarithmic form.

Transcript text: 8. Write the equation $5^{3 x}=71$ in logarithmic form.

Solution

Solution Steps

To convert the given exponential equation $5^{3x} = 71$ into logarithmic form, we need to use the property of logarithms that states $a^b = c$ can be written as $b = \log_a(c)$.

Solution Approach

The equation $5^{3x} = 71$ can be rewritten in logarithmic form as $3x = \log_5(71)$. To isolate $x$, we divide both sides by 3, resulting in $x = \frac{\log_5(71)}{3}$. Since most programming languages, including Python, use natural logarithms or base-10 logarithms, we can use the change of base formula: $\log_5(71) = \frac{\log(71)}{\log(5)}$.

Step 1: Convert to Logarithmic Form

Starting with the equation $5^{3x} = 71$, we can express it in logarithmic form: \[ 3x = \log_5(71) \]

Step 2: Apply Change of Base Formula

Using the change of base formula, we can rewrite $\log_5(71)$ as: \[ \log_5(71) = \frac{\log(71)}{\log(5)} \] Calculating this gives us: \[ \log_5(71) \approx 2.6486 \]

Step 3: Solve for $x$

Now, substituting back into the equation for $x$: \[ x = \frac{\log_5(71)}{3} \approx \frac{2.6486}{3} \approx 0.8829 \]