Solving Exponential Equations With Different Bases

How to solve exponential equations with different bases

When you have an exponential equation of the form a^x = b^y where a and b are different bases and cannot easily be converted to the same base, the most effective strategy is to take the logarithm of both sides of the equation.

You can use any base logarithm, but log₁₀ (common log) or ln (natural log) are typically preferred because they are readily available on calculators.

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

 

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Key Logarithm Property Needed
The crucial property that makes this work is the Power Rule of Logarithms:
log_b(M^p)=p log_b(M)

This rule allows you to bring the exponent down as a multiplier, turning the exponential equation into a more manageable algebraic equation.

Videos

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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