Solving Exponential Equations With Different Bases

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The following diagram shows the steps to solve exponential equations with different bases. Scroll down the page for more examples and solutions.

How to solve exponential equations with different bases?

Sometimes we are given exponential equations with different bases on the terms. In order to solve these equations we must know logarithms and how to use them with exponentiation. We can access variables within an exponent in exponential equations with different bases by using logarithms and the power rule of logarithms to get rid of the base and have just the exponent.

How to solve exponential equations using logarithms?

1. Isolate the exponential part of the equation. If there are two exponential parts put one on each side of the equation.
   
2. Take the logarithm of each side of the equation.
   
3. Solve for the variable.
   
4. Check your solution graphically.
   

Example:
Solve the exponential equations. Round to the hundredths if needed.
(a) 7^x - 1 = 4
(b) 3•2^x - 2 = 13
(c) (2/3)^x = 5^{3 - x}
(d) 5^{x - 3} = 3^{2x + 1}

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How to solve exponential equations with different bases?

When it’s not convenient to rewrite each side of an exponential equation so that it has the same base, you do the following:

1. Take the log (or ln) of both sides
   
2. Apply power property
   
3. Solve for the variable
   

Example:
Solve for x.
a) 6^x = 42
b) 7^x = 20
c) 8^{2x - 5} = 5^{x + 1}
d) 3^x = 5^{x - 1}

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How to use change of base formula to solve basic exponential equations?

Example:
Solve by writing as a log equation and then using the change of base formula. Round to 4 decimal places.
a) 5^x = 476
b) 4•3^x = 984

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How to solve exponential and logarithmic equations?

Principle used: log_aa^g(x) = g(x)

Examples:

1. Solve for x: 8^{8x + 7} = 9
   
2. Solve for x:
   a) 6^{1 - 2x} = 3
   b) 1 + 3(4^{2x + 3}) = 8
   
3. Solve for x: log₂4x + log₂x = 2
   
4. Solve for x:
   a) log₃5 + log₂(x + 4) = 2
   b) log₄x + log₄(x + 3) = 1

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How to solve exponential equations that are not one-to-one?

Examples:
3^{4x - 5} = 17

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