How to Solve Exponentials?

To solve exponential equations, the approach depends on whether the bases are the same or different. Here's a breakdown:

Solving Exponential Equations with the Same Base

When you have exponential equations with the same base, the simplest method is to equate the exponents. This works because if b^x = b^y, then x = y.

Steps:

1. Ensure both sides of the equation have the same base.
2. Set the exponents equal to each other.
3. Solve the resulting equation for the variable.

Example:

Solve for x: 2^x = 2⁵

Since the bases are the same (both are 2), we can set the exponents equal:

x = 5

Solving Exponential Equations with Different Bases

When the bases are different, you'll typically need to use logarithms.

Steps:

1. Apply a logarithm to both sides of the equation. You can use any base logarithm, but common choices are the common logarithm (base 10) or the natural logarithm (base e).
2. Use the power rule of logarithms to bring the exponent down as a coefficient. The power rule states that log_b(x^y) = y log_b(x).
3. Solve the resulting equation for the variable.

Example:

Solve for x: 3^x = 7

1. Apply the natural logarithm (ln) to both sides: ln(3^x) = ln(7)
2. Use the power rule: x ln(3) = ln(7)
3. Solve for x: x = ln(7) / ln(3)

You can use a calculator to find the approximate decimal value of x.

Using Logarithms Even with the Same Bases

While you can solve equations with the same bases by equating the exponents, you can also use logarithms. The method still works, although it's generally less efficient.

Example:

Solve for x: 2^x = 2⁵ (Using logarithms)

1. Apply the natural logarithm (ln) to both sides: ln(2^x) = ln(2⁵)
2. Use the power rule: x ln(2) = 5 ln(2)
3. Solve for x: x = (5 ln(2)) / ln(2) = 5