If you’re trying to solve the exponential equation 3^x × 9^x = 18, you’re in the right place. This step-by-step guide explains exactly how to solve for x using the laws of exponents. This type of algebra problem is common in high school mathematics, college algebra, competitive exams, and math olympiad preparation.
You can also watch the complete video solution below.
Problem
Solve:
3^x × 9^x = 18
At first glance, this equation looks difficult because it contains two exponential expressions. The key is to recognize that 9 is a power of 3.
Step 1: Rewrite everything using the same base
Since
9 = 3^2,
we can rewrite
9^x = (3^2)^x = 3^(2x).
The equation now becomes
3^x × 3^(2x) = 18
Step 2: Apply the law of exponents
When multiplying powers with the same base, add the exponents.
3^x × 3^(2x) = 3^(3x)
The equation is now
3^(3x) = 18
This is much easier to solve.
Step 3: Solve for x
Take the logarithm of both sides.
3x log(3) = log(18)
Divide both sides by 3 log(3).
x = log(18) / (3 log(3))
Using a calculator,
x ≈ 0.8797
Final Answer
x = log(18) / (3 log(3))
Approximate value:
x ≈ 0.8797
Why This Method Works
Many exponential equations become much simpler when every exponential expression is rewritten using the same base. Once the bases match, you can apply the exponent law
a^m × a^n = a^(m+n)
to combine the terms into a single exponential equation.
Whenever you see numbers like 4 and 16, 5 and 25, or 3 and 9 in the same equation, always check whether one number is a power of the other. This simple observation often makes the problem much easier.
Practice Similar Problems
Try solving these using the same strategy.
- 2^x × 4^x = 32
- 5^x × 25^x = 125
- 4^x × 16^x = 64
- 7^x × 49^x = 343
Watch More Exponential Equation Tutorials
If you enjoy solving exponential equations step by step, explore more tutorials covering exponent rules, logarithms, algebra techniques, and olympiad-style mathematics. New math problem breakdowns are published regularly, so be sure to subscribe and check back for more worked examples.