How to Solve Logarithms With Different Bases

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Logarithms are an important concept for the science and engineering world. A logarithm is the inverse of an exponent, much the same way addition is the inverse of subtraction. Logarithms provide an intuitive means of understanding multiplication by enabling a means of multiplying numbers using addition. Logarithms have a base, which is the number that is raised to some power for exponents. There are many operations that can be performed on logarithms; however, this requires that the logarithms have the same base. Solving logarithms with different bases require a change of base of the logarithms, which can be performed in a few short steps.

  • Write the question you are trying to solve. As an example, assume you are trying to solve the problem: log4(x + 1) + log16(x + 1) = log4(8). In this problem, there are two different bases: 4 and 16.

  • Use the change of base formula to give each term the same base. The change of base formula says that to change the base of logb(x), where b is the base and x is an arbitrary number, rewrite the logarithm as logk(x) / logk(b), where k is an arbitrary number selected as the new base. In the example above, you can change the base of the term log16(x + 1) by rewriting the number as log4(x + 1) / log4(16). This simplifes to log4(x + 1) / 2.

  • Use the rules of logarithms to simplify the problem into solvable form. In the example above, the equation log4(x + 1) + log4(x + 1) / 2 = log4(8) can be simplified to log4(x + 1) + log4(x + 1)^(1/2) = log4(8), using the power rule for logarithms. By using the product rule for logarithms, the equation can be further simplified to log4(x + 1)(x + 1)^(1/2) = log4(8).

  • Eliminate the logarithm. By taking both sides of the equation to the power of 4, the example equation simplifies to (x + 1)(x + 1)^(1/2) = 8, which further simplifies to (x + 1)^(3/2) = 8.

  • Solve for x. In the example above, this is done by taking both sides of the equation to the power of 2/3. This renders x + 1 = 4 and solving for x produces x = 3.

References

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