Solving Exponential Equations

What does it mean for a function to be exponential?

• A function is exponential if it can be expressed in the form @@f(x) = a imes b^x@@, where @@a@@ is a constant and @@b@@ is a positive real number.

• A function is exponential if it has a constant rate of change.

• A function is exponential if it can be represented as a linear equation.

• A function is exponential if it oscillates between two values.

How do you determine when an exponential function reaches a certain value?

• Set the function equal to the desired value and solve for the variable.

• Graph the function and find the intersection point.

• Differentiate the function and find the critical points.

• Use the quadratic formula to find the roots.

What is the value of @@e^0@@?

• 0

• 1

• e

• undefined

What is the significance of the base in an exponential function?

• The base determines the rate of growth or decay.

• The base has no effect on the function.

• The base only affects the horizontal shift of the graph.

• The base is always a positive integer.

How do you solve @@6e^{\frac{x}{3}} = 1@@?

• Divide both sides by 6, then take the natural logarithm: @@\frac{x}{3} = \text{ln}(\frac{1}{6})@@, and solve for @@x@@.

• Multiply both sides by 6, then take the square root: @@x = 6^{2}@@.

• Subtract 6 from both sides, then take the logarithm: @@\frac{x}{3} = \text{log}(1 - 6)@@, and solve for @@x@@.

• Add 6 to both sides, then take the cube root: @@x = 6^{3}@@.

How do you solve @@6^{3x} = 16@@?

• Take the logarithm of both sides: @@3x \text{ln}(6) = \text{ln}(16)@@, then solve for @@x@@.

• Multiply both sides by 3: @@3(6^{3x}) = 3(16)@@, then solve for @@x@@.

• Subtract 16 from both sides: @@6^{3x} - 16 = 0@@, then solve for @@x@@.

• Divide both sides by 6: @@\frac{6^{3x}}{6} = \frac{16}{6}@@, then solve for @@x@@.

What is an exponential equation?

• An equation where the variable is in the exponent, e.g., @@2^x = 8@@.

• An equation that involves only linear terms.

• An equation that has a constant term raised to a variable power.

• An equation that represents a straight line on a graph.

How do you solve an exponential equation with the same base?

• Set the bases equal to each other.

• Set the exponents equal to each other.

• Add the exponents together.

• Multiply the bases together.

What is the natural logarithm?

• The logarithm to the base 10.

• The logarithm to the base 2.

• The logarithm to the base @@e@@, where @@e@@ is approximately equal to 2.718.

• The logarithm of a number without a specified base.

How do you find the value of an exponential function at a given point?

• Substitute the value of @@x@@ into the function.

• Multiply the base by the exponent.

• Add the base and the exponent together.

• Differentiate the function with respect to @@x@@.