Solve Exponential Equations - Algebra-2

1. Fundamental Concepts

Definition: Exponential equations are equations where the variable appears in the exponent, such as $$a^x = b$$ .
Logarithmic Functions: The inverse of exponential functions, used to solve for variables in exponents, e.g., $$\log_a(b) = x \text{ if } a^x = b$$ .
Properties of Logarithms: Rules that help simplify and solve logarithmic expressions, including $$\log_a(xy) = \log_a(x) + \log_a(y)$$ and $$\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$$ .

2. Key Concepts

Basic Rule: $$a^m \cdot a^n = a^{m+n}$$

Solving Exponential Equations: If bases are the same, set exponents equal: $$a^x = a^y \Rightarrow x = y$$

Application: Used in various fields like finance (compound interest), biology (population growth), and physics (radioactive decay)

3. Examples

1. Easy

Question: Solve the equation $2^{x + 1}=8$

Solution:

Step 1: Rewrite the right-hand side of the equation as an exponential expression with the same base as the left-hand side. Since $8 = 2^3$ , the original equation can be transformed into $2^{x + 1}=2^3$ .

Step 2: According to the same-base method, set the exponents equal: $x + 1 = 3$ .

Step 3: Solve the integral equation to get $x = 2$ .

Verification: Substitute $x = 2$ into the original equation. The left-hand side is $2^{2 + 1}=2^3 = 8$ , and the right-hand side is $8$ . The equation holds, so $x = 2$ is the solution to the original equation.

2. Medium

Question: Solve the equation $3^{2x}=15$

Solution:

Step 1: The bases on both sides of the equation are different (3 and 15) and cannot be directly converted to the same base. Take the natural logarithm of both sides, resulting in $\ln 3^{2x}=\ln 15$ .

Step 2: Use the logarithmic property $\ln a^x = x\ln a$ to simplify the left-hand side to $2x\ln 3=\ln 15$ .

Step 3: Solve the linear equation in one variable $x$ to get $x=\frac{\ln 15}{2\ln 3}$ . Using the change-of-base formula, this can be further rewritten as $x=\frac{1}{2}\log_3 15$ .

3. Difficult

Question: Solve the equation $2^{x + 1}-3\times2^x + 2 = 0$

Solution:

Step 1: Observe the equation and find that it can be simplified using the substitution method. Let $t = 2^x$ (where $t>0$ , since the range of an exponential function is $(0,+\infty)$ ). Then $2^{x + 1}=2\times2^x = 2t$ , and the original equation can be transformed into $2t - 3t + 2 = 0$ .

Step 2: Simplify the integral equation to get $-t + 2 = 0$ , and solve for $t = 2$ .

Step 3: Substitute $t = 2$ back into the substitution expression $t = 2^x$ , resulting in $2^x = 2^1$ . According to the same-base method, solve for $x = 1$ .

Verification: Substitute $x = 1$ into the original equation.

The left-hand side is $2^{1 + 1}-3\times2^1 + 2 = 4 - 6 + 2 = 0$ , and the right-hand side is $0$ . The equation holds.

At the same time, it is necessary to confirm that $t = 2^1 = 2>0$ during substitution, which satisfies the domain requirement.

Therefore, $x = 1$ is the solution to the original equation.

4. Problem-Solving Techniques

Isolate the Exponential Term: Get the exponential term by itself on one side of the equation.
Use Logarithms: If the bases are not the same, take the logarithm of both sides to bring down the exponent.
Check Solutions: Always verify solutions by substituting them back into the original equation.