Exponential Equation with Unlike Bases 3^(1 - 2x) = 4^x

TL;DR

Learn how to solve exponential equations using logarithms, specifically using log base 3 in this example.

Transcript

solve 3 to the 1 minus 2x equals 4 to the x solution so we have an exponential equation and a lot of times when we have these types of equations we'll try to write both sides using the same base however this time we have a 3 and a 4 so that doesn't appear to be a very easy option so we have a couple choices but all of them really involve taking the... Read More

Key Insights

• ❓ Exponential equations with different bases can be solved using logarithms.
• 🧑‍💻 Log base 3 or log base 4 can be chosen to simplify the equation.
• ✊ The properties of logarithms, such as the power rule, can be used to manipulate the equation.
• 🥺 Rearranging the equation and isolating the variable leads to finding the solution.

Questions & Answers

Q: What method is used to solve exponential equations with different bases?

The content suggests using logarithms, specifically log base 3 or log base 4, to simplify the equation and make it easier to solve by using the properties of logarithms.

Q: Why is log base 3 chosen in this example instead of log base 4?

Log base 3 is chosen because it appears more complicated in the equation, providing an additional challenge to practice using logarithms. However, log base 4 can also be used as an alternative method.

Q: How is the equation rearranged to isolate the variable, x?

By adding 2x to both sides of the equation, all the x terms are combined to one side. This allows for factoring out x and dividing both sides by the remaining term (2 + log base 3 of 4) to solve for x.

Q: What is the final solution to the exponential equation?

The solution to the equation is x = 1 / (2 + log base 3 of 4).

Summary & Key Takeaways

• The content explains how to solve an exponential equation using logarithms.

• Log base 3 is used to simplify the equation and eliminate the different bases.

• The equation is rearranged to isolate the variable, x, and find the solution.