Solving exponential equation | Exponential and logarithmic functions | Algebra II | Khan Academy

TL;DR

The video explains how to solve an exponential equation by isolating the variable and using logarithms.

Transcript

Voiceover:Let's say that we've got the function y is equal to five times two to the t power. Someone were to come up to you and say, "Hey look, this is an interesting function." "But I'm curious, I like the number 1,111." "I'm curious at what point for what t value" "will my y be equal to 1,111." I encourage you to pause this video and think about ... Read More

Key Insights

• 😀 The function y = 5 * 2^t is used as an example to demonstrate solving exponential equations.
• ❓ Isolating the variable and applying logarithms are essential steps in solving exponential equations.
• 💨 Logarithms provide a way to invert an exponential function and find the exponent needed to obtain a given value.
• ⚾ Calculators with logarithmic functions can be used to compute logarithms, even if the desired base is not available.

Questions & Answers

Q: How can we solve the equation y = 5 * 2^t for a specific value of y?

To solve for a specific value of y, divide both sides of the equation by 5 to isolate the exponential term. Then, use logarithms to solve for t by taking the logarithm (base 2) of the resulting expression. Finally, calculate the logarithm (base 10) of the quotient and divide it by the logarithm (base 10) of 2 to find the value of t.

Q: What is the relationship between logarithms and exponential functions?

Logarithms and exponential functions are inverses of each other. If a^b = c, then log base a of c = b. Logarithms help us determine the exponent needed to obtain a certain value when using a specific base.

Q: Why is it necessary to use logarithms to solve exponential equations?

Logarithms help us isolate the exponent in an exponential equation, making it easier to find its value. By converting the exponential equation into a logarithmic equation, we can solve for the variable of interest.

Q: How do we compute logarithms that are not easily calculable?

If the desired logarithm does not have a readily available base on a calculator, we can use the change of base formula. This formula allows us to compute logarithms using a different base by taking the logarithm of the desired value divided by the logarithm of the new base.

Summary & Key Takeaways

• The video addresses finding the value of "t" in the equation y = 5 * 2^t that makes y equal to 1,111.

• By dividing both sides of the equation by 5, the equation is simplified to 2^t = 1,111/5.

• The use of logarithms is then introduced to solve for "t", and the equation becomes t = log base 2 of (1,111/5).

• To calculate this, the logarithm of (1,111/5) is divided by the logarithm of 2, resulting in an approximate value of t as 7.796.