Given that y=(3)4^x+² find the value of x when y=6

The equation given is \( y = 3 \cdot 4^x + 2 \). We need to find the value of \( x \) when \( y = 6 \).

Start by substituting \( y = 6 \) into the equation:

\[
6 = 3 \cdot 4^x + 2
\]

Next, subtract 2 from both sides to isolate the term involving \( x \):

\[
6 - 2 = 3 \cdot 4^x
\]

\[
4 = 3 \cdot 4^x
\]

Now, divide both sides by 3:

\[
\frac{4}{3} = 4^x
\]

To solve for \( x \), take the logarithm of both sides. Using the natural logarithm (ln):

\[
\ln\left(\frac{4}{3}\right) = \ln(4^x)
\]

Apply the logarithm power rule on the right side:

\[
\ln\left(\frac{4}{3}\right) = x \cdot \ln(4)
\]

Finally, solve for \( x \):

\[
x = \frac{\ln\left(\frac{4}{3}\right)}{\ln(4)}
\]

This is the exact solution. To get a numerical value, we can calculate it:

\[
x \approx \frac{\ln\left(\frac{4}{3}\right)}{\ln(4)} \approx \frac{\ln(1.3333)}{1.3863} \approx \frac{0.2877}{1.3863} \approx 0.2075
\]

So, \( x \approx 0.2075 \).

Would you like further details or have any questions?

Here are some related questions to consider:
1. How does the base of the logarithm affect the solution of an equation like this?
2. What if the equation was \( y = 3 \cdot 2^x + 2 \)? How would that change the value of \( x \) for \( y = 6 \)?
3. How do logarithmic properties simplify the process of solving exponential equations?
4. What if \( y = 3 \cdot 4^{2x} + 2 \)? How would the solution process differ?
5. Can this equation be solved graphically? If so, how?

**Tip:** When solving exponential equations, logarithms are a powerful tool to isolate the exponent and solve for the variable.

Learn how to solve the exponential equation 3 * 4^x + 2 = 6 step-by-step with logarithm properties and exponential function formulas. Suitable for Grades 10-12 students studying logarithms and exponential equations.

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