Questions: Solve for (x) : [ frac97 log 3 x=10 x= ] Solve for (x) : [ log 4(x)+log 4(x+1)=3 ] (x=)

Solve for (x) : [ frac97 log 3 x=10 x= ] Solve for (x) : [ log 4(x)+log 4(x+1)=3 ] (x=)
Transcript text: Solve for $x$ : \[ \begin{array}{l} \frac{9}{7} \log _{3} x=10 \\ x=\square \end{array} \] You may enter the exact value or round to 4 significant decimal places. Question Help: Video Message instructor Post to forum Submit Question Jump to Answer Question 13 Solve for $x$ : \[ \log _{4}(x)+\log _{4}(x+1)=3 \] Hint: When solving for x , you may need to use the quadratic formula. \[ x= \] $\square$ You may enter the exact solution or round to 4 decimal places.
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Solution

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Solution Steps

Step 1: Combine the Logarithms

Using the property $\log(M) + \log(N) = \log(M \cdot N)$, combine the logs: $\log_{4}((x + 0)(x + 1))$

Step 2: Isolate the Logarithm

Isolate the logarithm to get $\log_{4}(X) = 3$, where $X$ is the combined expression.

Step 3: Exponentiate

Exponentiate both sides to eliminate the logarithm: $4^{\log_{4}(X)} = 4^3$, simplifying to $X = 4^3$.

Step 4: Solve for $x$

Solve the equation $X = 4^3$ for $x$. This involves algebraic manipulation.

Step 5: Check Solutions

Valid solutions after checking the domain constraints: [7.516]

Final Answer:

$x = 7.516$

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