1+log3(x^4+25)=log√3(√30x^2+12)

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Jun 15, 2024

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Solving the Equation: 1 + log3(x^4 + 25) = log√3(√30x^2 + 12)

In this article, we will solve the equation 1 + log3(x^4 + 25) = log√3(√30x^2 + 12) using the properties of logarithms.

Step 1: Simplify the Right-Hand Side of the Equation

First, let's simplify the right-hand side of the equation using the property of logarithms:

loga(MN) = loga(M) + loga(N)

We can rewrite the right-hand side of the equation as:

log√3(√30x^2 + 12) = log√3(√30x^2) + log√3(12)

Step 2: Use the Property of Logarithms to Simplify Further

Next, we can use the property of logarithms:

loga(N^m) = m * loga(N)

to simplify the right-hand side of the equation further:

log√3(√30x^2) = 1/2 * log√3(30x^2)

Step 3: Simplify the Left-Hand Side of the Equation

Now, let's simplify the left-hand side of the equation:

1 + log3(x^4 + 25)

Using the property of logarithms:

loga(N) = loga(N) / loga(a)

we can rewrite the left-hand side of the equation as:

1 + log3(x^4 + 25) = 1 + log3(x^4) + log3(1 + 25/x^4)

Step 4: Equate the Two Expressions

Now, we can equate the two expressions:

1 + log3(x^4) + log3(1 + 25/x^4) = 1/2 * log√3(30x^2) + log√3(12)

Step 5: Solve for x

To solve for x, we can start by isolating the term log3(x^4) on the left-hand side of the equation:

log3(x^4) = 1/2 * log√3(30x^2) + log√3(12) - log3(1 + 25/x^4)

Using the property of logarithms:

loga(N^m) = m * loga(N)

we can rewrite the equation as:

4 * log3(x) = 1/2 * log√3(30x^2) + log√3(12) - log3(1 + 25/x^4)

Simplifying the equation further, we get:

log3(x) = 1/8 * log√3(30x^2) + 1/4 * log√3(12) - 1/4 * log3(1 + 25/x^4)

Finally, we can exponentiate both sides of the equation to solve for x:

x = 3^(1/8 * log√3(30x^2) + 1/4 * log√3(12) - 1/4 * log3(1 + 25/x^4))

The solution to the equation 1 + log3(x^4 + 25) = log√3(√30x^2 + 12) is x = 3^(1/8 * log√3(30x^2) + 1/4 * log√3(12) - 1/4 * log3(1 + 25/x^4)).

Conclusion

In this article, we solved the equation 1 + log3(x^4 + 25) = log√3(√30x^2 + 12) using the properties of logarithms. We simplified the equation step by step and finally solved for x. The solution to the equation is x = 3^(1/8 * log√3(30x^2) + 1/4 * log√3(12) - 1/4 * log3(1 + 25/x^4)).