Logarithms

Evaluating Logarithms Without a Calculator

100

log₂(4)

2

200

log₄(1/64)

-3

300

ln_7 17

approx 1.46

400

log_(1/2)9

approx -3.17

500

ln_(3/2)6

approx 4.42

Condense and Expand

100

log(x^6y^3)

6log(x)+3log(y)

200

3ln(x)-5ln(xy)

ln(1/(x^2y^5))

300

log root(3)(xyz)

log(x)/3+log(y)/3+log(z)/3

400

ln(sqrt(x^3-2)/x)

1/2ln(x^3-2)-ln(x)

500

2(log(2x)-log(y))-(log(3)+2log(5))

log((4x^2)/(75y^2))

Solving Logarithmic Equations

100

log₇(9x-4) = log₇(x+20)

x = 3

200

log_3 x=-4

x=1/81

300

log_x 16=2

x=pm4

400

log_2 (x-1)+log_2 (3)=5

x=35/3

500

log_3(x-1)-log_3(x+4)=log_3(5)

no solution

Transformations

100

3log(-x)

Vertical Stretch by 3

Reflect over the y-axis

200

-1/2log(x)+5

Reflect over the x-axis

Vertical compression of 1/2

Up 5 units

300

2/3log(-2x+7)-8

Vertical compression of 2/3

Reflect over the y-axis

Horizontal stretch of 2

Left 7 units

Down 8 units

400

log_3(x)+7

Up 7 units

Vertical Compression of 

1/ln(3)

500

log_(1/2)(x+4)-2

Reflect over the x-axis

Vertical compression of 

1/ln(2)

Left 4 units

Down 2 units

Solving Exponential Equations

100

4^(x-2)=64

x = 5

200

2^(x-5)=8^(x-3)

x = 2

300

5^(2-x)=3^(3x+1)

x approx 0.4322

400

7(1/3)^((2x)/3)=63

x=-3

500

(3^x-3^-x)/5=4

x = 2.73