求解指数方程$3^{x+2} + 2^{x+2} + 2^x = 2^{x+5} + 3^x$的方法咨询

API 探路者

2026-4-23

Hey Eddie, let's break down this exponential equation step by step—exponential equations can feel intimidating at first, but simplifying the terms will make this straightforward. Here's how to solve it:

Step 1: Expand all terms using exponent rules

First, use the rule $a^{m+n} = a^m \cdot a^n$ to rewrite every term so we have only $3^x$ and $2^x$ as the variable parts:

$3^{x+2} = 3^2 \cdot 3^x = 9 \cdot 3^x$
$2^{x+2} = 2^2 \cdot 2^x = 4 \cdot 2^x$
$2^{x+5} = 2^5 \cdot 2^x = 32 \cdot 2^x$

Substitute these back into the original equation:

$9 \cdot 3^x + 4 \cdot 2^x + 2^x = 32 \cdot 2^x + 3^x$

Step 2: Combine like terms

Let's group the $3^x$ terms on one side and $2^x$ terms on the other:

First, combine the $2^x$ terms on the left: $4 \cdot 2^x + 2^x = 5 \cdot 2^x$
Now move $3^x$ from the right to the left, and $5 \cdot 2^x$ from the left to the right:
$9 \cdot 3^x - 3^x = 32 \cdot 2^x - 5 \cdot 2^x$
Simplify both sides:
$8 \cdot 3^x = 27 \cdot 2^x$

Step 3: Isolate the ratio of the exponential terms

Divide both sides by $8 \cdot 2^x$ (since $2^x$ is always positive, we don't have to worry about division by zero):
(3^x)/(2^x) = 27/8

Notice that $(3^x)/(2x) = (3/2)^x$, and $27/8 = (3/2)^3$ (since $3^3=27$ and $2^3=8$). So now our equation becomes:
$(3/2)^x = (3/2)^3$

Step 4: Solve for x

Since the base $(3/2)$ is positive and not equal to 1, the only way the two exponential expressions are equal is if their exponents are equal. That gives us:
$x = 3$