Jeopardy - Exponential Equations & Logarithms

Exponential Equations

300

Solve for the exponential equation WITHOUT using logarithms or a calculator.

2^(3x)=64

x=2

Since 2⁶=64 we can rewrite the equation as 2^3x=2⁶.  That means 3x=6, so x=2.

400

Solve the exponential equation WITHOUT using logarithms or your calculator.

x=9

Since 27^x+3=(3³)^(x+3) we can rewrite the equation as 3^4x=3^3x+9.  That means 4x=3x+9, so x=9.

500

Find the exponential equation for the table below.

Logarithms

100

What is the OUTPUT of a LOGARITHMIC function?

an EXPONENT

Logarithms are the inverse of exponential functions and used when solving exponential equations where the exponent is unknown.  The output of a logarithm is an exponent.

200

Write the logarithm in EXPONENTIAL form.

7^-2=1/49

300

Estimate the value of log₃ (90) WITHOUT a calculator.

xapprox4.1

Since 3⁴=81 and 3⁵=243, we can see x must be between 4 and 5.  Since 90 is much closer to 81 than 243, we can estimate the value to be slightly over 4.

400

Solve for x  WITHOUT a calculator.

x=-2 and 4

First write the equation in exponential form.  Then use Algebra to isolate x.

2^3=(x^2-6x)/(1-x)

500

y=log_3((x-5)/2)-4

The inverse of an exponential function is a logarithm.  So rewrite the original function in logarithmic form by switching x and y and then isolating y.

Talk About It

100

What are the restrictions on the BASE of an EXPONENTIAL function?

The base can be any positive number except 1.

0<base, base ne1

200

Explain why the BASE of an exponential function can never be negative.

Thinking in terms of (1+R)...if the decay rate (R) is greater than one, it means you lost MORE than what you originally started with...not logical.

Thinking in terms of r^x...if you raise a negative base to an even power, the result is positive.  If you raise the same base to an odd power, the result is negative.  This would create a discrete graph like below...

300

What does the value of 2 in the exponential equation represent? Explain.

f(x)=2∙5^x

INITIAL VALUE

When the base of this function is raised to an input of 0, the result is 1.  So your are left with the coefficient being multiplied by 1.  This means the coefficient is the INITIAL VALUE or what you started with before the function began to grow or decay.

400

Does this exponential function model GROWTH or DECAY? Explain.

y=0.75(5/6)^x+3

DECAY

The common ratio (r) in this equation is less than 1 which indicated decay (the outputs are getting smaller by a factor of 5/6 for every increase of 1 in the input.