- WE WANT TO SOLVE 
THE EXPONENTIAL EQUATION.
WE WANT TO GIVE 
THE EXACT SOLUTION
WHICH WILL REQUIRE LOGARITHMS 
AND A DECIMAL APPROXIMATION
TO FOUR DECIMAL PLACES.
SO THERE ARE SEVERAL WAYS 
TO GO ABOUT SOLVING FOR X
BUT WHAT I'M GOING TO DO 
IS GET THE EXPONENTIAL PART
ON THE LEFT SIDE 
OF THE EQUATION
AND MOVE EVERYTHING ELSE 
ON THE RIGHT SIDE.
SO LET'S START 
BY DIVIDING BOTH SIDES BY 5.
THIS WILL GIVE US 1.05 
TO THE POWER OF X
EQUALS 7/5 TIMES 1.12 
RAISED TO THE POWER OF X.
NOW WE'LL DIVIDE BOTH SIDES 
BY 1.12 TO THE POWER OF X.
NOTICE ON THE RIGHT SIDE 
THIS SIMPLIFIES TO 1.
ON THE LEFT SIDE 
NOTICE HOW BOTH OF THESE BASES
ARE BEING RAISED 
TO THE POWER OF X
WHICH MEANS YOU CAN WRITE THIS 
AS A FRACTION
RAISED TO THE POWER OF X.
THIS WOULD BE 1.05 
DIVIDED BY 1.12
RAISED TO THE POWER OF X 
EQUALS 7/5.
AND NOW I WANT 
TO TAKE ADVANTAGE
OF THE POWER PROPERTY 
OF LOGARITHMS
SO I CAN MOVE THIS X 
OUT OF THE EXPONENT POSITION.
SO NOW I'LL TAKE 
THE COMMON LOG OR NATURAL LOG
OF BOTH SIDES OF THE EQUATION.
LET'S GO AHEAD AND USE 
THE COMMON LOG THIS TIME.
SO WE'D HAVE THE COMMON LOG 
OF 1.05 DIVIDED BY 1.12
RAISED TO THE POWER OF X 
EQUALS THE COMMON LOG OF 7/5.
NOW THE MAIN REASON 
WE WANTED TO DO THIS
BECAUSE NOW WE CAN APPLY THE 
POWER PROPERTY OF LOGARITHMS
WITH THIS PROPERTY HERE
SO WE CAN TAKE THIS EXPONENT 
AND MOVE IT TO THE FRONT
SO THAT WE HAVE X TIMES LOG OF 
THIS FRACTION EQUALS LOG 7/5.
AND NOW WE CAN SOLVE FOR X
BY DIVIDING BOTH SIDES 
OF THE EQUATION
BY THIS LOGARITHM HERE.
NOTICE ON THE LEFT SIDE 
OF THIS EQUATION
THIS SIMPLIFIES TO 1
SO WE'RE LEFT WITH X
EQUALS THE QUOTIENT 
OF THESE TWO LOGARITHMS HERE
WHICH WILL BE THE EXACT VALUE 
OF X.
SO WE HAVE A LOG 7/5 
DIVIDED BY THE LOG OF 1.0
DIVIDED BY 1.12.
SO THIS WILL BE ONE WAY TO 
EXPRESS THE EXACT VALUE OF X
THAT WOULD SATISFY THE GIVEN 
EXPONENTIAL EQUATION.
OF COURSE, WE COULD USE 
THE QUOTIENT PROPERTY
OF LOGARITHMS HERE
TO WRITE THE NUMERATOR HERE 
AS A DIFFERENCE OF TWO LOGS
AND THE SAME 
WITH THE DENOMINATOR
BUT THE VALUE 
WOULD BE THE SAME.
AND NOW TO GET 
A DECIMAL APPROXIMATION
WE'LL HAVE TO USE 
THE CALCULATOR.
SO WE'LL HAVE A SET OF 
PARENTHESES FOR THE NUMERATOR
AND A SET FOR THE DENOMINATOR.
SO WE'LL HAVE AN OPEN 
PARENTHESIS COMMON LOG 7
DIVIDED BY 5 CLOSE PARENTHESIS 
FOR THE LOGARITHM
AND CLOSE PARENTHESIS 
FOR THE NUMERATOR
DIVIDED BY COMMON LOG OF 1.05 
DIVIDED BY 1.12,
CLOSE PARENTHESIS 
FOR THE LOGARITHM
AND CLOSE PARENTHESIS 
FOR THE DENOMINATOR.
AFTER ROUNDING 
TO FOUR DECIMAL PLACES
X WILL BE APPROXIMATELY 
- 5.2135.
I HOPE THIS WAS HELPFUL.
