solve the equation. Round to the nearest thousandth 4^x=18

Question

1.solve the equation. Round to the nearest thousandth 4^x=18

 

2.use graphing utility to find the solution of the equation. round to the nearest thousandth e^x + 2= 1/x+4

 

3. solve for the indicated variable. logx (z+y)=logx(2x-4), for y

 

4. solve the equation f(x)=0 analytically and then use the graph of y=f(x) to solve the inequalities. f(x)<0 and f(x)greater than equal to 0. f(x)=ln (x+8)-ln (x-8) 

Answer 1

1)

 

4^x = 18

 

 

This is a logarithmic equation because we solving for the unknown exponent.

 

Log both sides of the equation and bring the exponent down as the coefficient of the log.

 

xlog(4) = log(18)

 

 

Divide both sides of the equation by log(4).

 

x = log(18) / log(4)

 

 

Your graphing calculator will have a log function to perform such calculations.

 

 

 

2)

 

On your calculator press the Y= key.

 

Type these equations into your calculator.

 

\Y₁ = e^x + 2

 

\Y₂ = 1 / x + 4

 

 

Press the GRAPH key.  You will see your graph being constructed.

 

Then press 2ND, TRACE, scroll down to select intersection.  Press Enter.  It will take you back to the graphing screen.  Press Enter 3 times to get your point of intersection.  Read the x value.  That is your solution.

 

 

3)

 

logx(z + y) = logx(2x - 4)

 

Since the logs are the same, we just equate their arguments.

 

z + y = 2x - 4

 

 

Solve for y,

 

y = 2x - 4 - z

 

 

 

4)

 

f(x) = ln(x + 8) - ln(x - 8)

 

 

According to the properties of logs.

 

f(x) = ln[(x + 8) / (x - 8)]

 

 

Set f(x) equal to zero.

 

0 = ln[(x + 8) / (x - 8)]

 

 

The solution to a natural log is the exponent of the base number e.

 

e⁰ = (x + 8) / (x - 8)

 

1 = (x + 8) / (x - 8)

 

 

Multiply both sides of the equation by (x - 8).

 

x - 8 = x + 8

 

 

The function has no solutions when f(x) equal zero.

 

For the inequality f(x) less than zero, use the graphing calculator.  Look for x values in which the graph is below the axis.

 

 

For the inequality f(x) greater then zero, look for x values in which the graph is above the x-axis.