What Is The Solution Of Log2 3x 7 3 One Third 4 5 Sixteen

Hannah Arendt

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29-11-2024

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This article will demonstrate how to solve the logarithmic equation log₂(3x + 7) = 3. We will utilize the definition of logarithms to convert the equation into an equivalent exponential form, and then proceed to solve for x.

Understanding the Logarithm

The equation log₂(3x + 7) = 3 means "to what power must we raise 2 to get 3x + 7?" The base of the logarithm is 2, and the result is 3.

Converting to Exponential Form

Recall that the logarithmic equation logₐ(b) = c is equivalent to the exponential equation aᶜ = b. Applying this to our equation:

log₂(3x + 7) = 3 is equivalent to 2³ = 3x + 7

Solving for x

Now we have a simple algebraic equation to solve:

2³ = 3x + 7

8 = 3x + 7

Subtract 7 from both sides:

8 - 7 = 3x

1 = 3x

Divide both sides by 3:

x = ⅓

Verifying the Solution

To confirm our solution, substitute x = ⅓ back into the original equation:

log₂(3(⅓) + 7) = log₂(1 + 7) = log₂(8) = 3

Since the equation holds true, our solution is correct.

Conclusion

Therefore, the solution to the logarithmic equation log₂(3x + 7) = 3 is x = ⅓. This problem highlights the importance of understanding the relationship between logarithmic and exponential forms, a fundamental concept in algebra and pre-calculus.