Solve for $$ x $$ $$ 10^{5 x-3}=9^{9 x-5} $$ $$ x=\square $$ You may enter the exact value or round to 4 decimal places. $$ $$ Next Question

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Solve for $$ x $$ $$ 10^{5 x-3}=9^{9 x-5} $$ $$ x=\square $$ You may enter the exact value or round to 4 decimal places. $$ > $$ Next Question

UpStudy AI · Accepted Answer

Solve the equation $$ 10^{5x-3}=9^{9x-5} $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$10^{5x-3}=9^{9x-5}$$
- step1: Take the logarithm of both sides:
  $$\log_{10}{\left(10^{5x-3}ight)}=\log_{10}{\left(9^{9x-5}ight)}$$
- step2: Evaluate the logarithm:
  $$5x-3=2\left(9x-5ight)	imes \log_{10}{\left(3ight)}$$
- step3: Calculate:
  $$5x-3=2\log_{10}{\left(3ight)}	imes \left(9x-5ight)$$
- step4: Expand the expression:
  $$5x-3=18\log_{10}{\left(3ight)}	imes x-10\log_{10}{\left(3ight)}$$
- step5: Move the expression to the left side:
  $$5x-18\log_{10}{\left(3ight)}	imes x=-10\log_{10}{\left(3ight)}+3$$
- step6: Add and subtract:
  $$\left(5-18\log_{10}{\left(3ight)}ight)x=-10\log_{10}{\left(3ight)}+3$$
- step7: Divide both sides:
  $$\frac{\left(5-18\log_{10}{\left(3ight)}ight)x}{5-18\log_{10}{\left(3ight)}}=\frac{-10\log_{10}{\left(3ight)}+3}{5-18\log_{10}{\left(3ight)}}$$
- step8: Divide the numbers:
  $$x=\log_{\frac{10^{5}}{3^{18}}}{\left(\frac{1000}{59049}ight)}$$
- step9: Simplify:
  $$x=3\log_{\frac{10^{5}}{3^{18}}}{\left(10ight)}-10\log_{\frac{10^{5}}{3^{18}}}{\left(3ight)}$$
The solution to the equation $$10^{5x-3}=9^{9x-5}$$ is $$x \approx 0.493624$$.
$$ x \approx 0.4936 $$

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