How to remove log from an equation?
Understand the Problem
The question is asking for a method or steps to isolate or eliminate a logarithmic term from an equation. This generally involves using properties of logarithms and exponents to restructure the equation into a solvable form.
Answer
Exponentiate both sides using the base of the logarithm.
Answer for screen readers
Use the property $\log_b(y) = x$ to rewrite it as $y = b^x$. For instance, $\log_b(x) = c$ can be rewritten as $x = b^c$.
Steps to Solve
- Identify the logarithmic term
Locate the logarithmic term in the equation that you want to remove.
- Use the inverse property
Recall that logarithms and exponentials are inverse operations. That is, if you have $\log_b(y) = x$, you can rewrite it as $y = b^x$.
Example: If $\log_b(x) = c$, you can rewrite it as $x = b^c$.
- Apply the inverse to both sides
To isolate the variable inside the logarithm, exponentiate both sides of the equation using the base of the logarithm.
Example: Given $\log_b(x) = c$, apply $b^{()}$ to both sides: $$b^{\log_b(x)} = b^c$$ Since $b^{\log_b(x)} = x$, we get: $$x = b^c$$
- Isolate the variable
Once the logarithm is removed, the equation should be in a simpler form where the variable is isolated or can be further simplified.
Example: If you start with $\log_b(x + 3) = 2$, exponentiate both sides: $$x + 3 = b^2$$ Then solve for $x$: $$x = b^2 - 3$$
Use the property $\log_b(y) = x$ to rewrite it as $y = b^x$. For instance, $\log_b(x) = c$ can be rewritten as $x = b^c$.
More Information
Logarithms and exponentials are known as inverse functions. They essentially 'undo' each other. This is a fundamental concept in mathematics.
Tips
A common mistake is not using the same base when applying the inverse property. Ensure that the base of the exponent corresponds exactly to the base of the logarithm you're trying to eliminate.
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