College Algebra Jeopardy

Exponent Rules

100

Simplify:

                           2³ · 2⁴ 

When multiplying exponential expressions with the same base, add the exponents!

            2³ · 2⁴ = 2³⁺⁴ = 2⁷ = 128

200

Simplify:

                          3⁵ / 3² 

When dividing exponential expressions with the same base, subtract the exponents!

             3⁵ / 3² = 3^5-2 = 3³ = 27

300

Simplify:

                           (x² · y³)⁴ 

Apply the power to each factor inside the parentheses!

        (x² · y³)⁴ = x^2x4 · y^3x4 = x⁸ · y¹²

400

Simplify:

                           (2³ · 4²)²

First, express 4 as 2²

    2³ · 4² = 2³ · (2²)² = 2³ · 2⁴ = 2³⁺⁴ =2⁷ 

Then raise to the power of 2

           (2⁷)² = 2^7x2 = 2¹⁴ = 16,384

500

Simplify:

                       (3x²)³ / (9x⁴)²

               (3x²)³ = 3³x^2x3 = 27x⁶

               (9x⁴)² = 9²x^4x2 = 81x⁸ 

Now divide: 

   27x⁶ / 81x⁸ = 27x^6-8/81 =1x^-2/3 = 1/3x²

Solving Exp. with Same Base

100

Solve for x: 

                      2^x = 2⁵   

Since the bases are the same, set the exponents equal!

                          x = 5

200

Solve for x:

                          3^x+1=3⁴

         

               x + 1 = 4 ----> x=3

300

Solve for x:

                         5^2x = 5⁸

                 2x = 8 ----> x = 4

400

Solve for x:

                         7^3x+1 = 7^x+7

    3x + 1 = x + 7 ---> 2x = 6 ---> x = 3

500

Solve for x:

                       2^2x / 2^x+3 = 2⁴

 2^{2x - (x+3)} = 2^x-3 = 2⁴ --> x−3 = 4 --> x = 7

Conversions Between Log and Exp.

100

        Write log₂ 8 = 3 in exponential form

                         2³ = 8

200

     Write 10⁴ = 10,000 in logarithmic form

                   log₁₀ 10,000 = 4. 

300

      Write log₅ 25 = 2 in exponential form

                         5² = 25

400

           Write 3^x = 81 in logarithmic form 

                       and solve for x

                      log₃ 81 = x   

Since 81 = 3⁴, then

                          x = 4

500

      Write log_b (x²) = 3 in exponential form

            and solve for x in terms of b

              b³ = x² ---> x = ±b^3/2 

Expanding Logarithms

100

Expand: 

                           log (xy)

              log (xy) = log x + log y

200

Expand:

                          log (x/y)

             log (x/y) = log x − log y

300

Expand:

                         log(x²y³) 

log(x²y³) = logx² + logy³ = 2 logx + 3 logy

400

Expand:

                       log (a³b² / c)

       log (a³b² / c) = loga³ + logb² − logc 

             = 3 loga + 2 logb − logc.

500

Expand:

                      log ((xy)⁵ / z³)

   log ((xy)⁵ / z³) = 5(logx + logy) − 3 logz 

             = 5 logx + 5 logy − 3 logz.

Condensing Logarithms

100

Condense:

                     log(x) + log(y)

             log x + log y = log(xy)

200

Condense:

                      log(x) − log(y)

             log x − log y = log (x / y)

300

Condense:

                    2log(x) + 3log(y)

   2logx + 3logy = logx² +logy³ = log(x²y³)

400

Condense:

            1/2 log(a) + log(b) − log(c)

                1/2 loga +logb − logc 

              = loga^1/2 + logb − logc 

              = log (a^1/2b / c)

           

500

Condense:

            log(x²) − 3log(y) + 1/2log(z)

              logx² − 3logy + 1/2logz 

           = 2logx − 3logy + logz^1/2 

           = log (x²√z / y³)