Exponential equations are essentially mathematical formulas that incorporate exponents. These equations usually take the shape of a^x = b, where 'a' denotes the base, 'x' symbolizes the exponent, and 'b' signifies the outcome. These types of equations are commonly utilized in simulating growth or decay processes, the dynamics of population, as well as calculations related to finance among other uses.
| Topic | Problem | Solution |
|---|---|---|
| None | An SUV is purchased new for $\$ 31,500$. (b) Sup… | The initial value of the SUV is given as \(V_{0} = \$ 31,500\). |
| None | A bacteria culture initially contains 1500 bacter… | This problem involves exponential growth, which can be represented by the formula \(P = P0 * e^{rt}… |
| None | Find the solution of the exponential equation \[ … | Divide both sides of the equation by 100 to get \(1.05^{2t} = 5000\) |
| None | Write the equation in exponential form. Assume th… | Write the equation in exponential form. Assume that all constants are positive and not equal to 1 . |
| None | A vehicle purchased for $\$ 29,800$ depreciates a… | The problem provides that a vehicle was purchased for $29,800 and depreciates at a constant rate of… |
| None | Solve the following equation for $x$ : \[ 2^{-x}=… | We are given the equation \(2^{-x}=15\). |
| None | Solve for $y$ in the equation below. Round your a… | Take the natural logarithm (ln) of both sides of the equation \(e^{y-3}=4\). This gives us \(y-3 = … |
| None | Solve the equation. \[ 2^{5 x+1}=4 \] The solutio… | The given equation is an exponential equation. To solve it, we need to express both sides of the eq… |
| None | Solve the exponential equation. Express irrationa… | Take the natural logarithm on both sides of the equation \(e^{x^{2}}=128\) to bring down the expone… |
| None | Find a formula for the exponential function passi… | The general form of an exponential function is \(f(x) = ab^x\). We can use the two given points to … |
| None | The exponential model $A=106.1 e^{0.02 t}$ descri… | The exponential model \(A=106.1 e^{0.02 t}\) describes the population, \(A\), of a country in milli… |
| None | The price $p$, in dollars, of a specific car that… | Given the function \(p(x)=22,265(0.89)^{x}\), where \(p\) is the price of the car in dollars and \(… |
| None | Solve the given exponential equation. Express the… | Divide both sides of the equation by 3 to isolate the exponential term: \(e^{x} = \frac{131}{3}\) |
| None | Question 14 (3 points) Han purchased a car for $\… | The depreciation of the car can be modeled using an exponential decay formula. The general formula … |
| None | Solve the following exponential equation. Express… | Take the natural logarithm on both sides of the equation to bring down the exponent and solve for x… |
| None | Use a calculator with a $y^{\mathrm{x}}$ key or a… | The exponential function \(f(x)=540(1.032)^{x}\) models the population of a country, \(f(x)\), in m… |
| None | $a^{x}+b^{y}=c^{z}$ | Given the equation: \(a^{x}+b^{y}=c^{z}\) |
| None | $2^{2 x}+4 \cdot 2^{x}=-4$ | Let \(y = 2^x\), then the equation becomes \(y^2 + 4y = -4\) |
| None | Work out the value of $k$ if \[ k \times 3^{-2}=4… | Rewrite the equation as: \(k = \frac{4}{3^{-2}}\) |
| None | Solve the exponential equation for \( x \). \[ 9^… | 9^{3x-10}=(81^{-1})^{\frac{1}{6}} |
| None | \( 7^{3 x-6}+100=101 \) | \( 7^{3 x-6} = 1 \) |
| None | \( \begin{array}{l}5^{x}+5^{x}+5^{x}+5^{x}+5^{x}=… | \(5^{x}+5^{x}+5^{x}+5^{x}+5^{x}=5^{x+1}\) |
| None | 6. Since 1992, the value of homes in a neighborho… | a. Since 2000 is 8 years after 1992, divide 16 years by 2 to get the half-growth period: \(\frac{16… |
| None | \( x^{3} \cdot x^{\stackrel{2}{?}}=\frac{1}{x^{12… | \( x^3 \cdot x^y = x^{3+y} \) |