Questions: Match each equation with a graph above: 10^x a. green (G) e^x b. black (K) log (x) c. red (R) ln (x) d. blue (B)

Transcript text: Match each equation with a graph above: $10^{x}$ a. green (G) $e^{x}$ b. black (K) $\log (x)$ c. red (R) $\ln (x)$ d. blue (B)

Solution

Solution Steps

Step 1: Analyze the graphs and equations

We are given four graphs (green, black, red, blue) and four equations ($10^x$, $e^x$, $\log(x)$, $\ln(x)$). We need to match the graphs with the corresponding equations.

Step 2: Exponential functions

The equations $10^x$ and $e^x$ represent exponential functions. Their graphs are increasing curves that pass through the point (0,1). Since $e \approx 2.718 < 10$, the graph of $e^x$ grows slower than $10^x$. Looking at the provided image, the blue (B) curve is steeper than the green (G) curve. Therefore, the blue (B) graph represents $10^x$, and the green (G) graph represents $e^x$.

Step 3: Logarithmic functions

The equations $\log(x)$ and $\ln(x)$ represent logarithmic functions. Their graphs are increasing curves that pass through the point (1,0). Since $\log(x)$ is the logarithm base 10 and $\ln(x)$ is the logarithm base $e$, and $e < 10$, the graph of $\ln(x)$ grows faster than $\log(x)$. Looking at the provided image, the black (K) curve is steeper than the red (R) curve. Therefore, the black (K) graph represents $\ln(x)$, and the red (R) graph represents $\log(x)$.

Final Answer

$10^x$: d. blue (B) $e^x$: a. green (G) $\log(x)$: c. red (R) $\ln(x)$: b. black (K) \$ \boxed{ \begin{aligned} 10^x &: \text{ d. blue (B)} \\ e^x &: \text{ a. green (G)} \\ \log(x) &: \text{ c. red (R)} \\ \ln(x) &: \text{ b. black (K)} \end{aligned} } \$

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