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How do you solve #3^(x-2)=13^(4x)#?

Answer 1

Madelyn Anderson

Take natural logarithm both side

#ln(3^(x-2))=ln(13^(4x))#

#xln(3)-2ln(3) = 4xln(13)#

#2ln(3)=xln(3)-4xln(13)#

#2ln(3)=x(ln(3)-4ln(13))#

#(2ln(3))/(ln(3)-4ln(13)) = x#

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Answer 2

Serenity Johnson

To solve (3^{x-2} = 13^{4x}), we can rewrite (13^{4x}) as ((3^2)^{4x}), since (13 = 3^2).

So, the equation becomes:

[3^{x-2} = (3^2)^{4x}]

Using the properties of exponents, we can simplify ((3^2)^{4x}) to (3^{2 \cdot 4x}), which equals (3^{8x}).

Now the equation becomes:

[3^{x-2} = 3^{8x}]

Since the bases are the same, we can equate the exponents:

[x - 2 = 8x]

Solve for (x):

[x - 8x = 2]

[-7x = 2]

[x = \frac{2}{-7}]

So, (x = -\frac{2}{7}).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.