To solve the equation $3^x - 4 = 6$ using the change of base formula, we will proceed step-by-step:

Step 1: Isolate the Exponential Expression

Start by isolating the term involving the exponential expression:

$3^x - 4 = 6$

Add 4 to both sides:

$3^x = 10$

Step 2: Apply Logarithm to Both Sides

To solve for $x$, take the logarithm of both sides. We can use any logarithm, but for consistency, let's use the natural logarithm (base $e$):

$\ln(3^x) = \ln(10)$

Step 3: Use the Logarithm Power Rule

Apply the power rule of logarithms, which states $\ln(a^b) = b \ln(a)$:

$x \ln(3) = \ln(10)$

Step 4: Solve for $x$

Now, solve for $x$ by dividing both sides by $\ln(3)$:

$x = \frac{\ln(10)}{\ln(3)}$

Step 5: Use the Change of Base Formula

The change of base formula for logarithms states that:

$\log_b(y) = \frac{\log_k(y)}{\log_k(b)}$

In this case, using natural logarithms ($\ln$) or common logarithms ($\log$) provides the same solution. Therefore:

$x = \frac{\log(10)}{\log(3)}$

Step 6: Calculate the Value of $x$

Now, compute the value using a calculator:

$x \approx \frac{\log(10)}{\log(3)} \approx \frac{1}{0.4771} \approx 2.096$

Step 7: Round to the Nearest Thousandth

Rounded to the nearest thousandth, the value of $x$ is:

$x \approx 2.096$

Final Answer

The solution for $x$ is approximately $2.096$.

Would you like more details or have any questions?

Related Questions:

1. How do you solve exponential equations that involve different bases?
2. Can you solve $2^x + 5 = 13$ using a logarithm?
3. What is the power rule of logarithms, and how is it applied?
4. How does the change of base formula work for logarithms other than base 10 or $e$?
5. What are the differences between natural logarithms ($\ln$) and common logarithms ($\log$)?

Tip:

When solving equations involving exponents, always consider using logarithms to simplify and isolate the variable.