How can I solve more complex exponential equations?

  • MHB
  • Thread starter mathdad
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In summary, the conversation discussed the ability to solve equations with one exponential term, but the equation presented with two exponential terms may require more advanced techniques such as numeric root-finding. The approximate solution for x was mentioned to be 2.5421632382360203811.
  • #1
mathdad
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I can solve equations like 4^(x) = 16 or
5^(x + 1) = 25. However, there are exponential equations that a bit more involved. The following equation has two exponentials on the left side.

Solve for x.

5^(x - 2) + 8^(x) = 200
 
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  • #2
RTCNTC said:
I can solve equations like 4^(x) = 16 or
5^(x + 1) = 25. However, there are exponential equations that a bit more involved. The following equation has two exponentials on the left side.

Solve for x.

5^(x - 2) + 8^(x) = 200

I don't believe you can solve that algebraically...I would use a numeric root-finding technique, such as the Newton-Raphson method, to approximate the solution to the desired number of decimal places:

\(\displaystyle x\approx2.5421632382360203811\)
 
  • #3
MarkFL said:
I don't believe you can solve that algebraically...I would use a numeric root-finding technique, such as the Newton-Raphson method, to approximate the solution to the desired number of decimal places:

\(\displaystyle x\approx2.5421632382360203811\)

Ok. Good to know. I don't feel so bad now.
 

FAQ: How can I solve more complex exponential equations?

What is an exponential equation?

An exponential equation is a mathematical expression in the form of y = ab^x, where a is a constant, b is the base, and x is the exponent. It is used to represent situations where a quantity grows or decays at a constant rate.

What are some real-life applications of exponential equations?

Exponential equations are used to model population growth, radioactive decay, compound interest, and the spread of infectious diseases. They can also be used to predict trends in data, such as stock market trends and sales growth.

How do you solve an exponential equation?

To solve an exponential equation, you can use logarithms or algebraic methods. If the equation is in the form of y = ab^x, you can take the logarithm of both sides to eliminate the exponent and then solve for x. If the equation is in the form of ab^x = c, you can take the logarithm of both sides and use the properties of logarithms to solve for x.

What is the difference between exponential and linear equations?

The main difference between exponential and linear equations is that the rate of change in exponential equations is constant, while the rate of change in linear equations is variable. In other words, the graph of an exponential equation appears as a curved line, while the graph of a linear equation appears as a straight line.

Why are exponential equations important in science?

Exponential equations are important in science because they allow us to model and understand natural phenomena that involve growth or decay at a constant rate. They also allow us to make predictions and projections based on data. Exponential equations are used in many fields of science, including biology, physics, chemistry, and economics.

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