Math  /  Algebra

QuestionSolve the equation: 52x=205^{2 x}=20
ln(20)2ln(5)\frac{\ln (20)}{2 \ln (5)} ln(10)ln(5)\frac{\ln (10)}{\ln (5)} ln(20)ln(5)\frac{\ln (20)}{\ln (5)} ln(4)2\frac{\ln (4)}{2}

Studdy Solution

STEP 1

1. The equation 52x=20 5^{2x} = 20 is exponential.
2. We will use logarithms to solve for x x .

STEP 2

1. Take the logarithm of both sides.
2. Use logarithmic identities to simplify.
3. Solve for x x .

STEP 3

Take the natural logarithm of both sides of the equation:
ln(52x)=ln(20) \ln(5^{2x}) = \ln(20)

STEP 4

Use the logarithmic identity ln(ab)=bln(a)\ln(a^b) = b \ln(a) to simplify the left side:
2xln(5)=ln(20) 2x \cdot \ln(5) = \ln(20)

STEP 5

Solve for x x by dividing both sides by 2ln(5) 2 \ln(5) :
x=ln(20)2ln(5) x = \frac{\ln(20)}{2 \ln(5)}
The value of x x is:
ln(20)2ln(5) \boxed{\frac{\ln(20)}{2 \ln(5)}}

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