General Method for Solving Exponential Equations

To solve an exponential equation, such as $a^x = b$, where $a$ and $b$ are constants, follow these steps:

1. Identify the Equation: Make sure the equation is in the form $a^x = b$.

2. Take the Logarithm of Both Sides: You can use natural logarithms (ln) or common logarithms (log). The choice depends on your preference or the context of the problem.

   $\ln(a^x) = \ln(b)$

3. Apply the Power Rule of Logarithms: This rule states that $\ln(a^b) = b \ln(a)$. Use this rule to bring the exponent $x$ down as a coefficient:

   $x \ln(a) = \ln(b)$

4. Solve for $x$: Isolate $x$ by dividing both sides of the equation by $\ln(a)$:

   $x = \frac{\ln(b)}{\ln(a)}$

Example

Consider the equation $2^x = 10$:

1. Take the Natural Logarithm of Both Sides: $\ln(2^x) = \ln(10)$

2. Apply the Power Rule: $x \ln(2) = \ln(10)$

3. Solve for $x$: $x = \frac{\ln(10)}{\ln(2)}$

Using a calculator:

• $\ln(10) \approx 2.3026$
• $\ln(2) \approx 0.6931$

So, $x \approx \frac{2.3026}{0.6931} \approx 3.3219$

Thus, the solution to $2^x = 10$ is approximately $x = 3.3219$.

Would you like more details or have any questions?

Here are some related questions to explore:

1. How do you solve exponential equations when the base is $e$ (Euler's number)?
2. What are the properties of logarithms that are particularly useful for solving equations?
3. Can we solve $5^x = 25$ without using logarithms?
4. How do we handle exponential equations with different bases, like $2^x = 3^{2x-1}$?
5. What is the significance of the natural logarithm in calculus?
6. How can we solve logarithmic equations?
7. What are some common applications of exponential and logarithmic functions in science and engineering?
8. How do we solve exponential inequalities, such as $2^x > 10$?

Tip: Familiarize yourself with the properties of logarithms, as they are essential for solving a wide range of mathematical problems, particularly those involving exponential functions.