Solve the equation $\frac{11}{7} \log _4(x)=10$ for $x$. You must enter your answer exactly (using rational powers). Reminder: you are not allowed to use a calculator.
Answer (2)
Isolate the logarithm by multiplying both sides of the equation by 11 7 .
Rewrite the equation in exponential form.
The solution is x = 4 11 70 .
The final answer is 4 11 70 .
Explanation
Problem Setup We are given the equation 7 11 lo g 4 ( x ) = 10 and we need to solve for x .
Isolating the Logarithm First, we want to isolate the logarithm. To do this, we multiply both sides of the equation by 11 7 : 11 7 ⋅ 7 11 lo g 4 ( x ) = 11 7 ⋅ 10 lo g 4 ( x ) = 11 70
Converting to Exponential Form Now, we rewrite the equation in exponential form. Recall that lo g b ( a ) = c is equivalent to b c = a . Applying this to our equation, we get: x = 4 11 70
Final Answer Thus, the solution for x is 4 11 70 .
Examples
Logarithmic equations are useful in many fields, such as calculating the magnitude of earthquakes on the Richter scale, determining the pH of a solution in chemistry, or modeling population growth in biology. For example, if we know the intensity of an earthquake is 1000 times greater than the smallest detectable wave, we can use logarithms to find its magnitude on the Richter scale.
To solve the equation 7 11 lo g 4 ( x ) = 10 , we isolate the logarithm to get lo g 4 ( x ) = 11 70 and then convert it to exponential form, yielding x = 4 11 70 .
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