How do you solve the equation $$8^x - 1 = 5$$ ?

To solve $$8^x - 1 = 5$$ , first add 1 to both sides to get $$8^x = 6$$ . Then take the logarithm of both sides: $$x = \frac{\text{log}(6)}{\text{log}(8)}$$ , which approximates to 0.86.

How do you express $$9^x = 4$$ in logarithmic form?

The equation $$9^x = 4$$ can be expressed in logarithmic form as $$x = \frac{\text{log}(4)}{\text{log}(9)}$$ .

What is the solution to the equation $$2 \times 3^x = 12$$ ?

To solve $$2 \times 3^x = 12$$ , divide both sides by 2 to get $$3^x = 6$$ . Then take the logarithm: $$x = \frac{\text{log}(6)}{\text{log}(3)}$$ , which approximates to 1.63.

How do you solve $$\frac{e^{(4x-3)}}{e^{(x-2)}} = 10$$ ?

To solve $$\frac{e^{(4x-3)}}{e^{(x-2)}} = 10$$ , simplify to $$e^{(4x-3) - (x-2)} = 10$$ , which leads to $$e^{(3x - 1)} = 10$$ . Taking the natural logarithm gives $$3x - 1 = \text{ln}(10)$$ , so $$x = \frac{\text{ln}(10) + 1}{3}$$ , which approximates to 1.10.

What is the natural logarithm?

The natural logarithm, denoted as $$\text{ln}$$ , is the logarithm to the base $$e$$ , where $$e$$ is approximately equal to 2.718.

What does it mean to round to the nearest hundredth?

Rounding to the nearest hundredth means to round a number to two decimal places. For example, 0.856 rounded to the nearest hundredth is 0.86.

What is the inverse operation of exponentiation?

The inverse operation of exponentiation is taking the logarithm. For example, if $$y = a^x$$ , then $$x = \text{log}_a(y)$$ .

How do you convert an exponential equation to a logarithmic equation?

To convert an exponential equation $$a^b = c$$ to logarithmic form, write $$b = \text{log}_a(c)$$ .

What is the value of $$\text{log}_2(8)$$ ?

The value of $$\text{log}_2(8)$$ is 3, because $$2^3 = 8$$ .

What is the exponential growth formula?

The exponential growth formula is $$N(t) = N_0 e^{rt}$$ , where $$N_0$$ is the initial amount, $$r$$ is the growth rate, and $$t$$ is time.

What is the difference between exponential growth and decay?

Exponential growth occurs when a quantity increases at a rate proportional to its current value, while exponential decay occurs when it decreases at a rate proportional to its current value.

How do you find the x-intercept of an exponential function?

To find the x-intercept of an exponential function, set the function equal to zero and solve for $$x$$ . However, exponential functions never cross the x-axis.

HPC 5.7 Day 1 Exponential Equations Dec 20 2024 Flashcards

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15 questions

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1.

FLASHCARD QUESTION

Front

What is an exponential equation?

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2.

FLASHCARD QUESTION

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3.

FLASHCARD QUESTION

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What is the base of an exponential function?

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4.

FLASHCARD QUESTION

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5.

FLASHCARD QUESTION

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6.

FLASHCARD QUESTION

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This property states that when multiplying two exponential expressions with the same base, you can add the exponents.

7.

FLASHCARD QUESTION

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