Sure! Let's solve the equation \( y = 3 \cdot 4^x + 2 \) using the common logarithm (base 10). 

Given that \( y = 6 \), we substitute this value into the equation:

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

Subtract 2 from both sides:

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

Divide both sides by 3:

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

Now, take the common logarithm (log base 10) of both sides:

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

Using the logarithm power rule, which states that \(\log(a^b) = b \cdot \log(a)\), we get:

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

Finally, solve for \( x \):

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

Let's calculate this using common logarithms:

\[
x = \frac{\log(1.3333)}{\log(4)}
\]

Using a calculator:

\[
\log(1.3333) \approx 0.1250 \quad \text{and} \quad \log(4) \approx 0.6021
\]

Thus,

\[
x \approx \frac{0.1250}{0.6021} \approx 0.2076
\]

So, \( x \approx 0.2076 \).

Would you like further details or have any questions?

Here are some related questions to consider:
1. How do logarithms simplify solving exponential equations?
2. What is the difference between using natural logarithms and common logarithms?
3. How can you verify this solution graphically?
4. What happens if the base of the exponent is a different number, like 10?
5. How would the equation change if we were solving for \( x \) in \( y = 5 \cdot 4^x + 2 \)?

**Tip:** Familiarity with logarithm properties, such as the power rule and change of base formula, can greatly simplify the process of solving exponential equations.

Learn how to solve exponential equations using common logarithms (base 10). Step-by-step guide with a detailed solution example involving the equation y = 3 * 4^x + 2. Suitable for high school students studying logarithms and exponential functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation