- WE WANT TO SOLVE 
THE GIVEN LOG EQUATION,
AND, IF NEEDED, 
ROUND TO FOUR DECIMAL PLACES.
NOTICE HOW IN THIS EQUATION,
THE TWO LOGARITHMS DON'T HAVE 
THE BASE LISTED
WHICH MEANS THESE ARE 
COMMON LOG OR LOG BASE 10.
WHEN WE HAVE AN EQUATION 
LIKE THIS
WHERE WE HAVE TWO LOGARITHMS,
OUR GOAL IS TO COMBINE 
THE LOGARITHMS
AND THEN WRITE 
THE LOG EQUATION
AS AN EXPONENTIAL EQUATION 
TO SOLVE FOR X.
SO TO COMBINE 
THESE TWO LOGARITHMS,
BECAUSE WE HAVE A SUM,
WE'LL USE THE PRODUCT PROPERTY 
OF LOGARITHMS GIVEN HERE
WHERE WE CAN COMBINE 
THE TWO LOGARITHMS
INTO A SINGLE LOG
IF WE MULTIPLY THE NUMBER 
PARTS OF THE LOGARITHMS.
SO COMMON LOG X + COMMON LOG 
OF THE QUANTITY X + 4
IS EQUAL TO THE COMMON LOG 
OF X x X + 4.
THIS IS STILL EQUAL TO +2.
LET'S GO AHEAD 
AND FIND THIS PRODUCT.
SO WE'LL HAVE LOG OF THE 
QUANTITY X SQUARED + 4X = 2,
AND AGAIN, REMEMBERING THIS 
IS COMMON LOG OR LOG BASE 10,
WE CAN NOW WRITE THIS 
AS AN EXPONENTIAL EQUATION.
SO STARTING WITH OUR BASE AND 
WORKING AROUND THE EQUAL SIGN,
WE CAN WRITE 
THE EXPONENTIAL EQUATION.
WE HAVE 10 RAISED 
TO THE 2ND POWER
MUST EQUAL THE QUANTITY X 
SQUARED + 4X.
SO 10 IS THE BASE, 
2 IS THE EXPONENT,
AND THE NUMBER 
IS X SQUARED + 4X.
SO AGAIN, 10 TO THE 2ND
MUST EQUAL THE QUANTITY 
X SQUARED + 4X.
10 SQUARED IS EQUAL TO 100.
NOTICE HOW WE HAVE TO SOLVE 
A QUADRATIC EQUATION,
SO LET'S GO AHEAD 
AND SET THIS EQUAL TO ZERO,
SO WE'LL SUBTRACT 100 
ON BOTH SIDES
WHICH WILL GIVE US ZERO = 
X SQUARED PLUS 4X - 100.
NOW, UNFORTUNATELY, 
THERE ARE NO FACTORS OF -100
THAT ADD TO +4,
WHICH MEANS TO SOLVE FOR X
WE'LL HAVE TO USE 
THE QUADRATIC FORMULA.
WHEN USING THE QUADRATIC 
FORMULA,
WE DO NEED TO REMEMBER THAT 
A IS GOING TO BE EQUAL TO +1,
B IS EQUAL TO +4, 
AND C IS EQUAL TO -100.
SO HERE'S 
THE QUADRATIC FORMULA.
SO NO WE'LL PERFORM 
SUBSTITUTION FOR A, B, AND C.
SO WE'LL HAVE X = -B OR -4 =/- 
THE SQUARE ROOT OF B SQUARED
OR 4 SQUARED - 4 x A 
WHICH IS 1 x C WHICH IS -100,
AND THIS IS ALL DIVIDED 
BY 2 x A OR 2 x 1.
NOW, WE'LL BEGIN TO SIMPLIFY.
WE'LL HAVE X = -4 +/- 
THIS IS GOING TO BE 16 + 400,
SO IT'S THE SQUARE ROOT OF 416 
DIVIDED BY 2.
LET'S CONTINUE SIMPLIFYING 
THIS ON THE NEXT SLIDE.
THE NEXT STEP IS TO SIMPLIFY 
THE SQUARE ROOT OF 416.
I'VE ALREADY MADE A PRIME 
FACTORIZATION TREE OVER HERE
ON THE RIGHT 
TO SAVE SOME TIME.
SO WE'LL HAVE X = -4 +/- 
THE SQUARE ROOT OF 416
BUT THE PRIME FACTORIZATION 
OF 416
HAS 1, 2, 3, 4, 5 FACTORS OF 2 
AND A FACTOR OF 13.
THIS IS STILL DIVIDED BY 2,
SO EVERY TIME WE HAVE 
TWO EQUAL FACTORS
WE HAVE A PERFECT SQUARE 
FACTOR.
SO THIS WILL SIMPLIFY, 
THIS WILL SIMPLIFY.
SO WE HAVE X = -4 +/-,
THIS IS GOING TO BE 4 SQUARE 
ROOT 26 ALL DIVIDED BY 2.
LET'S BREAK THIS UP 
INTO TWO FRACTIONS.
WE HAVE -4 DIVIDED BY 2 +/- 4 
SQUARE ROOT 26 DIVIDED BY 2.
SO WE HAVE X = -2 +/- 
THIS WILL BE 2 SQUARE ROOT 26.
BUT WE'RE STILL NOT DONE.
IF WE GO BACK 
TO THE ORIGINAL EQUATION,
NOTICE HOW THE VALUE OF X 
HAS TO BE GREATER THAN ZERO,
AND THIS REPRESENTS 
TWO SOLUTIONS.
ONE SOLUTION IS X = -2 + 2 
SQUARE ROOT 26,
AND THE OTHER SOLUTION 
IS X = -2 -2 SQUARE ROOT 26.
THIS VALUE HERE IS DEFINITELY 
GOING TO BE NEGATIVE.
SO THIS IS NOT A SOLUTION 
TO THE LOG EQUATION.
THIS IS THE EXACT SOLUTION 
TO OUR LOG EQUATION,
BUT LET'S ALSO FIND 
THE DECIMAL APPROXIMATION.
SO WE'LL ENTER THIS 
INTO THE CALCULATOR.
WE HAVE -2 + 2 SQUARE ROOT 26.
SO THE DECIMAL APPROXIMATION 
FOR X IS 8.1980.
SO DEPENDING 
ON WHETHER YOU'RE ASKED
TO FIND THE EXACT SOLUTION,
OR THE SOLUTION ROUNDED 
TO FOUR DECIMAL PLACES
WILL AFFECT HOW YOU GIVE 
YOUR ANSWER.
I HOPE YOU FOUND THIS HELPFUL.
