Question
upstudy study bank question image url

Solve the following exponential equations, giving the final answers correct to 3 significant figures. i. \( 5^{2 x-1}=4^{300} \) \( x \approx \) ii. \( 2^{y+1}=\frac{10}{2^{y}} \). \( y \approx \)

Ask by Hardy Smith. in Ghana
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

\( x \approx 129 \) and \( y \approx 0.349 \).

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

The Deep Dive

To solve the first equation \( 5^{2x-1} = 4^{300} \), we can take logarithms on both sides. 1. Taking logarithm: \[ \log(5^{2x-1}) = \log(4^{300}) \] 2. Applying the power rule: \[ (2x - 1) \log(5) = 300 \log(4) \] 3. Solving for \( x \): \[ 2x - 1 = \frac{300 \log(4)}{\log(5)} \] \[ 2x = \frac{300 \log(4)}{\log(5)} + 1 \] \[ x = \frac{1}{2} \left( \frac{300 \log(4)}{\log(5)} + 1 \right) \] Calculating \( x \) gives approximately \( x \approx 121.363 \), so to three significant figures, \( x \approx 121 \). For the second equation \( 2^{y+1} = \frac{10}{2^{y}} \): 1. Multiplying both sides by \( 2^{y} \): \[ 2^{y+1} \cdot 2^{y} = 10 \] 2. Simplifying: \[ 2^{2y + 1} = 10 \] 3. Taking logarithm: \[ (2y + 1) \log(2) = \log(10) \] 4. Solving for \( y \): \[ 2y + 1 = \frac{\log(10)}{\log(2)} \] \[ 2y = \frac{\log(10)}{\log(2)} - 1 \] \[ y = \frac{1}{2} \left( \frac{\log(10)}{\log(2)} - 1 \right) \] Calculating \( y \) gives approximately \( y \approx 0.5 \), so to three significant figures, \( y \approx 0.500 \).

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy