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Exponential Function Solver

Solve a^x = b and evaluate exponential functions

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Solve a^x = b for x
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Exponential Function Properties

f(x) = a^x, a>0, a!=1: domain R, range (0,inf)
a>1: increasing, 0<a<1: decreasing
a^x = b -> x = ln(b)/ln(a)
a^x * a^y = a^(x+y), (a^x)^y = a^(xy)

Exponential functions model growth and decay. To solve a^x=b, take the natural logarithm of both sides. The power rule brings down the exponent. Divide by ln(a) to isolate x. For real solutions, both a>0 and b>0 are required.

⚠ The base a must be positive (a>0) and not equal to 1. The result b must be positive for real solutions. Negative b gives complex solutions.

What Are Exponential Functions?

Exponential functions have the form f(x)=a^x where a is a positive constant. They grow (a>1) or decay (0<a<1) rapidly. The inverse is the logarithm. Key properties include a^0=1, positive range, and horizontal asymptote at y=0.

Definition

f(x)=a^x, a>0, a!=1. Always positive. f(0)=1. Horizontal asymptote y=0. Increasing if a>1.

Solving a^x=b

Check if b is power of a (integer solution). Otherwise: x=ln(b)/ln(a). Use log of any base. b>0 required.

Properties

a^x*a^y=a^(x+y), a^x/a^y=a^(x-y), (a^x)^y=a^(xy). a^(-x)=1/a^x. a^(1/n)=nth root of a.

Applications

Population growth, radioactive decay, compound interest, cooling, epidemic spread, and many natural phenomena.

Teaching Example: Solve 2^x=8. Since 8=2^3, same base: x=3. Using logs: x=ln(8)/ln(2)=2.0794/0.6931=3.0. Verify: 2^3=8.

Applications

Growth/Decay Finance Biology Physics Chemistry

Frequently Asked Questions

Solve a^x=b?▼
Take ln both sides: x*ln(a)=ln(b) -> x=ln(b)/ln(a). Check integer solutions: does b = a^n?
Exponential properties?▼
a^x*a^y=a^(x+y), (a^x)^y=a^(xy), a^0=1, a^(-x)=1/a^x. Always positive.
What if a=1?▼
1^x = 1 for all x. Not a proper exponential function. Cannot solve 1^x = b uniquely.
Negative base?▼
Negative bases give complex results for non-integer exponents. Standard exponential functions require a>0.

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