Exponential equations
An exponential equation can look intimidating until both sides share a base. Once they do, the exponents are equal and the rest is ordinary algebra. The same pattern sits behind growth and decay: the size of the factor tells you whether a quantity is rising or falling.
Rewrite to a common base
Many exponential equations fall apart once both sides share a base. If 3^(x + 1) = 27, write 27 as 3 cubed, so x + 1 = 3 and x = 2. Recognising powers of 2, 3, 5 and 10 quickly is the whole skill, because once the bases match the equation is linear.
Growth and decay factors
An expression like 250(1.08)^t grows by 8 percent each period, because the factor sits above one. A factor of 0.92 decays by 8 percent, because it sits below one. The number out front is the starting amount, and the exponent counts how many periods have passed.
Halving and doubling
Doubling uses a factor of 2 and halving uses 0.5. If a quantity doubles every 6 hours, the model is written with t divided by 6 in the exponent. Check the direction with a test value: after one full period the result should be twice the start, not half of it.
When the base does not line up
Not every base cooperates. If 2^x = 10, ten is not a clean power of two, so the answer is not a tidy integer. Questions are usually written so the bases do match, which is a hint to look for that rewrite before reaching for anything heavier.
How do I solve an exponential equation?
Rewrite both sides as powers of the same base. Once the bases match, set the exponents equal and solve the resulting equation.
What does the factor tell me?
A factor above one means growth, and a factor below one means decay. The distance from one gives the percent change per period.
What does a factor of 0.5 mean?
The quantity halves each period. After each step it is half of what it was before.
What if the bases cannot be matched?
Then the answer is not a clean integer. On the SAT most equations are built so the bases do line up, so look for that first.
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