Question
upstudy study bank question image url

Solve for \( x \) : \( 6^{4 x-4}=2^{5 x-10} \) \( x=\square \) You may enter the exact value or round to 4 decimal places.

Ask by Adkins Salazar. in the United States
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

\( x \approx 0.0637 \)

Solution

We start with the equation: \[ 6^{4x-4}=2^{5x-10}. \] First, notice that we can express \(6\) as \(2\cdot 3\). Doing so gives: \[ (2\cdot 3)^{4x-4}=2^{5x-10}. \] This can be rewritten as: \[ 2^{4x-4}\cdot 3^{4x-4}=2^{5x-10}. \] Now, divide both sides by \(2^{4x-4}\) (which is valid since \(2^{4x-4}\ne0\)): \[ 3^{4x-4}=2^{5x-10-(4x-4)}. \] Simplify the exponent on the right: \[ 5x-10-(4x-4)= (5x-4x) + (-10+4)= x-6. \] Thus, the equation becomes: \[ 3^{4x-4} = 2^{x-6}. \] Take the natural logarithm of both sides: \[ \ln\left(3^{4x-4}\right)=\ln\left(2^{x-6}\right). \] Using the logarithm power rule \(\ln(a^b)=b\ln(a)\), we have: \[ (4x-4)\ln3=(x-6)\ln2. \] Expand and rearrange the equation: \[ 4x\ln3 - 4\ln3 = x\ln2 - 6\ln2. \] Bring terms involving \(x\) to one side and constant terms to the other: \[ 4x\ln3 - x\ln2 = 4\ln3 - 6\ln2. \] Factor out \(x\) on the left-hand side: \[ x(4\ln3-\ln2)=4\ln3-6\ln2. \] Now solve for \(x\) by dividing both sides by \((4\ln3-\ln2)\): \[ x=\frac{4\ln3-6\ln2}{4\ln3-\ln2}. \] Thus, the solution is: \[ \boxed{x=\frac{4\ln3-6\ln2}{4\ln3-\ln2}}. \] To give a numerical approximation, we use: \[ \ln3\approx1.0986 \quad \text{and} \quad \ln2\approx0.6931. \] Calculating the numerator: \[ 4\ln3-6\ln2\approx4(1.0986)-6(0.6931)=4.3944-4.1586=0.2358, \] and the denominator: \[ 4\ln3-\ln2\approx4(1.0986)-0.6931=4.3944-0.6931=3.7013. \] Then, \[ x\approx\frac{0.2358}{3.7013}\approx0.0637. \] So the final answer is: \[ \boxed{x=\frac{4\ln3-6\ln2}{4\ln3-\ln2}\approx0.0637}. \]

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

The Deep Dive

To solve for \( x \) in the equation \( 6^{4x-4} = 2^{5x-10} \), we start by expressing both sides in terms of base 2. First, note that \( 6 = 2 \cdot 3 \), so: \[ 6^{4x-4} = (2 \cdot 3)^{4x-4} = 2^{4x-4} \cdot 3^{4x-4} \] Now we can rewrite the equation: \[ 2^{4x-4} \cdot 3^{4x-4} = 2^{5x-10} \] Dividing both sides by \( 2^{4x-4} \) gives us: \[ 3^{4x-4} = 2^{5x-10-4x+4} = 2^{x-6} \] Next, we take logarithms to solve for \( x \): \[ \log(3^{4x-4}) = \log(2^{x-6}) \] Using the logarithmic identity \( \log(a^b) = b\log(a) \), we rewrite this as: \[ (4x-4) \log 3 = (x-6) \log 2 \] Expanding both sides yields: \[ 4x \log 3 - 4 \log 3 = x \log 2 - 6 \log 2 \] Rearranging gives: \[ 4x \log 3 - x \log 2 = -6 \log 2 + 4 \log 3 \] Factoring \( x \) out from the left side: \[ x (4 \log 3 - \log 2) = 4 \log 3 - 6 \log 2 \] Now, solving for \( x \): \[ x = \frac{4 \log 3 - 6 \log 2}{4 \log 3 - \log 2} \] Using logarithmic approximations, where \( \log 2 \approx 0.3010 \) and \( \log 3 \approx 0.4771 \): \[ x = \frac{4 \cdot 0.4771 - 6 \cdot 0.3010}{4 \cdot 0.4771 - 0.3010} \] \[ x = \frac{1.9084 - 1.8060}{1.9084 - 0.3010} \approx \frac{0.1024}{1.6074} \approx 0.0637 \] Thus, rounding to four decimal places, we have: \[ x \approx 0.0637 \]

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