How to solve the equation 13^(3x-1)=15?the result is expected to have up to 4 decimals

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Wiggin42's profile pic

Wiggin42 | Student, Undergraduate | (Level 2) Valedictorian

Posted on

13^(3x-1)=15

Since the variable is in the exponent we have to do the opposite of an exponent. Much like how addition is the opposite of subtraction, logs are the opposite of exponents. 

Take the log of both sides, when you do this, the exponent comes down. 

(3x-1)log13 = 15

Now you can solve for x, keeping in mind that log13 is just a number so you can treat it like any other number. 

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giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

To solve for x, we'll have to take natural logarithms both sides:

ln 13^(3x-1) = ln 15

We'll use the power rule of logarithms:

(3x-1)*ln 13 = ln 15

We'll divide by ln 13 both sides of the equation:

(3x-1) = ln 15 /ln 13

We'll add 1  both sides:

3x = ln 15 /ln 13 + 1

We'll divide by 3:

x = (ln 15 /ln 13 + 1)/3

x = (ln 15 + ln 13)/3*ln 13

x = ln (15*13)/ln 13^3

We'll evaluate the numerator of the fraction:

ln (15*13) = 5.2729

We'll evaluate the denominator of the fraction:

ln 13^3 = 7.6948

x = 5.2729/7.6948

x = 0.6852

The requested soution of the equation, rounded up to 4 decimals, is x = 0.6852.

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