Podcast
Questions and Answers
When solving exponential equations graphically, what does the intersection of the graphs representing each side of the equation indicate?
When solving exponential equations graphically, what does the intersection of the graphs representing each side of the equation indicate?
- The equation has no solution.
- The equation is an identity and is true for all values of x.
- The x-coordinate of the intersection point(s) represent the solution(s) to the equation. (correct)
- The y-coordinate of the intersection point(s) represent the solution(s) to the equation.
Consider the equation (4^{x} = 2^{x-3}). What is the next logical step in solving the equation?
Consider the equation (4^{x} = 2^{x-3}). What is the next logical step in solving the equation?
- Graph each side of the equation and find the intersection.
- Divide both sides of the equation by 2.
- Take the logarithm of both sides of the equation.
- Rewrite both sides of the equation with the same base. (correct)
How does the property of equality for exponential equations simplify solving equations with the same base?
How does the property of equality for exponential equations simplify solving equations with the same base?
- It allows you to cancel out the bases, simplifying the equation.
- It allows you to equate the bases directly and solve for the variable.
- It allows you to take the logarithm of both sides and solve.
- It allows you to equate the exponents and solve for the variable. (correct)
Under what condition can you directly apply the property of equality for exponential equations?
Under what condition can you directly apply the property of equality for exponential equations?
In solving an exponential equation graphically, if the graphs of the two sides of the equation do not intersect, what does this imply?
In solving an exponential equation graphically, if the graphs of the two sides of the equation do not intersect, what does this imply?
If a population of bacteria is modeled by the equation (p = 30(2^t)), where (p) is the population and (t) is the time in years, how does increasing (t) affect the population?
If a population of bacteria is modeled by the equation (p = 30(2^t)), where (p) is the population and (t) is the time in years, how does increasing (t) affect the population?
Which method is most suitable for solving an exponential equation such as (5^{x} = 125)?
Which method is most suitable for solving an exponential equation such as (5^{x} = 125)?
What is the first step in solving (4^{x+1} = rac{1}{64})?
What is the first step in solving (4^{x+1} = rac{1}{64})?
In the context of exponential equations, what does it mean for an equation to have 'no solution'?
In the context of exponential equations, what does it mean for an equation to have 'no solution'?
When solving (9^{x+2} = 27^{x}), what base should both sides of the equation be rewritten with?
When solving (9^{x+2} = 27^{x}), what base should both sides of the equation be rewritten with?
Flashcards
Exponential equation
Exponential equation
Equations where variable expressions occur as exponents.
Equality for Exponential Equations
Equality for Exponential Equations
If b > 0 and b ≠ 1, then bx = by if and only if x = y.
Solving Exponential Equations
Solving Exponential Equations
Rewrite so each side has the same base.
Solving by Graphing
Solving by Graphing
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Exponential Equations
Exponential Equations
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Solving Exponential Equations
Solving Exponential Equations
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Study Notes
Solving Exponential Equations
- Exponential equations contain variable expressions as exponents.
- These can be solved by using the same base or graphing.
Property of Equality for Exponential Equations
- If powers have the same positive base (excluding 1), they are only equal if their exponents are equal.
- If b > 0 and b ≠ 1, then b^x = b^y if and only if x = y.
Solving Exponential Equations with the Same Base
- Rewrite each side of the equation using the same base, if possible
- Set the exponents equal to each other
- Reduce to find the value of x
- Solve: 3^(x+1) = 3^5 resulting in x = 4
- Solve: 6 = 6^(2x-3) resulting in x = 2
- Solve: 10^(3x) = 10^(2x+3) resulting in x = 3
Solving Exponential Equations with Unlike Bases
- Rewrite each side of the equation with the same base.
- Apply power properties where necessary, then set exponential terms equal to each other
- 5^x = 125 simplifies to 5^x = 5^3 so x = 3.
- 4^x = 2^(x-3) simplifies to (2^2)^x = 2^(x-3), then 2^(2x) = 2^(x-3) so 2x = x-3 and x = -3.
- 9^(x+2) = 27^x simplifies to (3^2)^(x+2) = (3^3)^x, then 3^(2x+4) = 3^(3x) so 2x+4 = 3x and x = 4.
Solving Exponential Equations When 0 < b < 1
- Rewrite fractions to have the same base as the whole number
- Equate exponents to solve
- (1/2)^x = 4 becomes (2^(-1))^x = 2^2, yielding -x = 2, and thus x = -2.
- 4^(x+1) = 1/64 becomes 4^(x+1) = 4^(-3) so x + 1 = -3 and x = -4
Solving Exponential Equations by Graphing
- Graph exponential equations to find the x value
- Use the "intersect" feature to determine the values in the graphs
- Some exponential equations have one solution, no solution, or multiple, depending on the number of intersections
- For the equation (1/2)^(x-1) = 7, graph y=(12)^(x-1) and y = 7, the intersect is approximately (-1.81, 7), so x ≈ -1.81
- For the equation 3^(x+2) = x + 1, graph y=3^(x+2) and y = x + 1, since the equations do not intersect, there is no solution
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