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How to Solve a Log Equation When Its Base Isn't 10

Logarithms and their partners, exponential functions, appear throughout equations in science and economics. A logarithm is the inverse function of an exponential, and as such, it shares a variety of characteristics, making it handy for numerically solving equations that would otherwise be cumbersome to solve. Although logarithms appear in thousands of different equations, often with different bases, a few simple guidelines can help you find a solution.

Things You'll Need

  • Means to perform calculations
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Instructions

    • 1

      Apply the inverse function where it is applicable.

      For example, if log_a (x) = y, then you would convert the equation to one in which each side is a power of a. That is, a^*log_a(x) = a^y, and because exponentiation and logarithms are inverse operations, this can be further simplified to x = a^y.

    • 2

      Use the rules of logarithms to simplify an expression where all the logs are with respect to the same base.

      For example, 2*log_a(y) = log_a(x+1) + log_a(4) can be rewritten as log_a(y^2) = log_a(4*(x+1)). You can then use each side as the exponent of the base a, and since exponentiation and logarithms are inverse operations, this results in the equation y^2 = 4*(x+1).

    • 3

      Use the base conversion formula to express all the logarithms in the equation with respect to the base 10. This approach is useful primarily when wanting to use a calculator or program to calculate a number.

      For example, 3*log_a(y) = log_b(7), can be rewritten as log_a(y^3) = log_10(7)/log_10(b). Because the logarithm to the base 10 is so common, it is usually written without including in the reference to the base, so the equation can be written as log_a(y^3) = log (7)/log(b). Using a calculator, this equation becomes log_a(y^3) = 0.845/log(b).

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