Example Question - converting to exponential form
Here are examples of questions we've helped users solve.
Solving Logarithmic Equations by Converting to Exponential Form
To solve the given logarithmic equation \(\log_4(5x + 9) = 3\), you can begin by converting the logarithmic form to exponential form. The basic logarithm rule states that if \(\log_b(A) = C\), then \(b^C = A\). Applying this rule here gives: \(4^3 = 5x + 9\) Now, calculate the left side: \(4^3 = 4 \times 4 \times 4 = 64\) So the equation becomes: \(64 = 5x + 9\) Now, subtract 9 from both sides to isolate the term containing \(x\): \(64 - 9 = 5x\) \(55 = 5x\) Finally, divide both sides by 5 to solve for \(x\): \(x = 55 / 5\) \(x = 11\) Therefore, the solution to the equation is \(x = 11\).
Solving Logarithmic Equations by Converting to Exponential Form
To solve the logarithmic equation \(\log(2x + 4) = 2\), we can rewrite the equation in its exponential form. The base of the logarithm is 10 by default when no base is specified. With this in mind, the equation becomes: \(10^{\log(2x + 4)} = 10^2\) Since \(10^{\log(x)} = x\), we have: \(2x + 4 = 10^2\) \(2x + 4 = 100\) Next, we solve for \(x\) by isolating the variable: \(2x = 100 - 4\) \(2x = 96\) Divide both sides by 2 to find \(x\): \(x = \frac{96}{2}\) \(x = 48\) Hence, the solution to the equation \(\log(2x + 4) = 2\) is \(x = 48\).
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