Solving Exponential and Logarithmic Equations
Master key strategies and techniques for solving equations with exponentials and logarithms.
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Study Smart Tips for Solving Exponential and Logarithmic Equations
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Questions Covered in This Set
8 cards to master
What is the first step when solving an exponential equation like 3^x = 50?
Take the logarithm of both sides (using ln or log), then use the power rule to bring the exponent down: x·ln(3) = ln(50).
How do you solve a logarithmic equation like log₂(x) = 5?
Rewrite in exponential form: x = 2^5 = 32. Use exponentiation as the inverse operation of logarithms.
Why must you always check solutions to logarithmic equations?
Logarithms require positive arguments. Algebraic solutions might produce negative values that make the original equation undefined.
For log(x) + log(x - 3) = 1, what property helps combine the terms?
The product rule: log(x) + log(x - 3) = log[x(x - 3)], which simplifies the equation before solving.
What substitution simplifies e^(2x) - 5e^x + 6 = 0?
Let u = e^x, so e^(2x) = u². The equation becomes u² - 5u + 6 = 0, which can be factored and solved.
If 5^(2x) = 5^(x+3), how can you solve for x directly?
Use the one-to-one property: if bases are equal and expressions are equal, exponents must be equal. So 2x = x + 3, giving x = 3.
What is ln(50)/ln(3) in the context of solving 3^x = 50?
It is the exact solution for x (approximately 3.56). This form is obtained after taking ln of both sides and isolating x.
In compound interest A = P(1 + r/n)^(nt), how do you solve for time t when A and P are known?
Divide both sides by P, then take ln of both sides to bring down the exponent, and isolate t: t = ln(A/P)/(n·ln(1 + r/n)).
