- WE WANT TO SOLVE 
THE GIVEN EXPONENTIAL EQUATION
USING LOGARITHMS.
WE'RE ALSO TOLD TO ROUND TO 
FOUR DECIMAL PLACES AS NEEDED.
TO SOLVE AN EXPONENTIAL EQUATION 
LIKE THIS
THE FIRST STEP IS TO ISOLATE 
THE EXPONENTIAL PART
OR ISOLATE 1.25 
RAISED TO THE POWER OF 8T.
WHICH MEANS THE FIRST STEP 
IS TO DIVIDE BOTH SIDES BY 25.
ON THE LEFT SIDE 25 DIVIDED BY 
25 SIMPLIFIES TO 1.
SO WE HAVE 1.25 RAISED 
TO THE POWER OF 8T EQUALS 40,000
DIVIDED BY 25 IS EQUAL TO 1,600.
NOW, TO SOLVE FOR T
WE WANT TO MOVE 8T OUT OF THE 
POSITION OF THE EXPONENT.
TO DO THIS WE CAN TAKE 
THE COMMON LOG OR NATURAL LOG
OF BOTH SIDES OF THE EQUATION.
WE NEED TO USE COMMON LOG 
OR NATURAL LOG
BECAUSE WE ARE GOING TO HAVE TO 
USE THE CALCULATOR.
LET'S GO AHEAD 
AND TAKE THE NATURAL LOG
OF BOTH SIDES OF THE EQUATION.
SO WE HAVE NATURAL LOG OF 1.25 
RAISED TO THE POWER OF 8T
EQUALS NATURAL LOG 1,600.
NOW LOOKING AT THE LEFT SIDE
WE CAN USE THE POWER PROPERTY 
OF LOGARITHMS
TO TAKE THE EXPONENT OF 8T
AND MOVE IT TO THE FRONT 
OF THE LOGARITHM
SO WE HAVE 8T 
TIMES NATURAL LOG 1.25.
SO AGAIN WE HAVE 8T TIMES 
NATURAL LOG 1.25
EQUALS NATURAL LOG 1,600.
NOW, REMEMBER OUR GOAL 
IS TO SOLVE FOR T
SO I AND NATURAL LOG 1.25 ARE 
ATTACHED TO T BY MULTIPLICATION
WHICH MEANS NOW WE'LL DIVIDE 
BOTH SIDES
BY 8 NATURAL LOG 1.25.
AND OF COURSE WE'LL DO THIS 
ON BOTH SIDES.
LOOKING AT THE LEFT SIDE 
8 DIVIDED BY 8 IS EQUAL TO 1.
NATURAL LOG 1.25 
DIVIDED BY NATURAL LOG 1.25
SIMPLIFIES TO 1.
SO WE'RE LEFT WITH T.
AND THEN WE'LL HAVE TO GO 
TO THE CALCULATOR
TO GET A DECIMAL APPROXIMATION 
FOR THIS QUOTIENT HERE.
SO I'LL PUT THE NUMERATOR 
IN THE SET OF PARENTHESES
AS WELL AS THE DENOMINATOR.
SO THE NUMERATOR IS GOING TO BE 
NATURAL LOG 1,600.
THIS PARENTHESIS 
IS FOR THE NATURAL LOG.
THIS PARENTHESIS 
IS FOR THE NUMERATOR
DIVIDED BY 8 NATURAL LOG 1.25.
AGAIN, ONE PARENTHESIS 
FOR THE NATURAL LOG
AND ONE FOR THE DENOMINATOR.
ROUNDING TO FOUR DECIMAL PLACES
WE CAN SEE T IS GOING TO BE 
APPROXIMATELY 4.1329.
SO THIS WOULD BE OUR SOLUTION
BUT IF WE WANT TO CHECK THIS
WE NEED TO GO BACK UP 
TO THE ORIGINAL EQUATION.
THIS IS TELLING US THAT 
25 TIMES 1.25
RAISED TO THE POWER OF 8 TIMES T 
WHICH IS APPROXIMATELY 4.1329,
IS APPROXIMATELY EQUAL TO 
40,000.
IT'S GOING TO BE APPROXIMATELY 
EQUAL TO 40,000
BECAUSE WE DO HAVE TO 
ROUND THIS VALUE
AND THEREFORE IT IS GOING TO BE 
OFF A LITTLE BIT.
LETS GO BACK TO THE CALCULATOR 
AND CHECK THIS.
25 TIMES 1.25 RAISED TO THE 
POWER OF 8 TIMES 4.1329.
NOTICE HOW IT IS APPROXIMATELY 
EQUAL TO 40,000.
THIS IS A LITTLE BIT LARGER
BECAUSE WE DID ROUND 
THE EXPONENT UP.
BUT THIS DOES VERIFY 
THAT OUR WORK IS CORRECT.
I HOPE YOU FOUND THIS HELPFUL.
