Question

name:
lt#5 i can solve exponential functions
solve the equation.

1. $4^x = 32$

1. $10^x = 10^{2x + 1}$

1. $2^x = 8^{x + 1}$

lt#6 i can write and graph exponential functions

1. does the table represent a linear function, exponential function, or neither?

explain.

x	0	1	2	3	4
y	1	3	9	27	81

1. graph $y = 2^x$. state the domain and range

Explanation:

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LT#5: Solve the exponential equations

Step1: Rewrite bases as powers of 2

$4^x = 32 \implies (2^2)^x = 2^5$

Step2: Simplify exponents

$2^{2x} = 2^5 \implies 2x = 5$

Step3: Solve for x

$x = \frac{5}{2}$

Step1: Set exponents equal (same base)

$10^x = 10^{2x+1} \implies x = 2x + 1$

Step2: Isolate x

$x - 2x = 1 \implies -x = 1 \implies x = -1$

Step1: Rewrite 8 as power of 2

$2^x = 8^{x+1} \implies 2^x = (2^3)^{x+1}$

Step2: Simplify exponents

$2^x = 2^{3(x+1)} \implies x = 3x + 3$

Step3: Isolate x

$x - 3x = 3 \implies -2x = 3 \implies x = -\frac{3}{2}$

1. For the table: A linear function has a constant rate of change (difference in y-values), while an exponential function has a constant ratio of y-values. Calculate the ratios: $\frac{3}{1}=3$, $\frac{9}{3}=3$, $\frac{27}{9}=3$, $\frac{81}{27}=3$. The ratio is constant, so it is exponential.
2. For $y=2^x$: The domain is all real numbers (no restrictions on x). The range is all positive real numbers, since $2^x$ is always positive and approaches 0 as $x\to-\infty$, and increases to $\infty$ as $x\to\infty$. To graph, plot key points: $(-1, \frac{1}{2})$, $(0,1)$, $(1,2)$, $(2,4)$, then draw a smooth curve through them, approaching the x-axis (y=0) as x decreases.

Answer:

1. $x = \frac{5}{2}$
2. $x = -1$
3. $x = -\frac{3}{2}$

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LT#6: Write and graph exponential functions