Logarithmic Scale Graphs

Understanding Logarithmic Scales

Logarithmic scales are used when data spans several orders of magnitude, or when a non-linear relationship can be transformed into a linear one by taking logarithms. This makes it easier to visualize trends and determine the constants in relationships like power laws or exponential functions.

Log-Linear Paper/Plot: One axis (typically the y-axis) has a logarithmic scale, and the other has a linear scale. This is useful for plotting exponential relationships like $y = ab^x$ or $y = ae^{kx}$. When plotted on log-linear scales (or after transforming $y$ to $\log(y)$ or $\ln(y)$ and plotting against $x$), these relationships become straight lines.
Log-Log Paper/Plot: Both axes have logarithmic scales. This is useful for plotting power law relationships like $y = ax^n$. When plotted on log-log scales (or after transforming both $y$ to $\log(y)$ and $x$ to $\log(x)$), these relationships become straight lines.

By transforming the data and plotting it, if a straight line is obtained, we can use linear regression to find the slope and intercept of this line. These values can then be used to determine the constants of the original non-linear equation.

Plot & Analyze Logarithmic Relationships

Power Law: $y = ax^n$ (Log-Log Transformation)

This relationship becomes linear when $\log_{10}(y)$ is plotted against $\log_{10}(x)$.

$$ \log_{10}(y) = n \cdot \log_{10}(x) + \log_{10}(a) $$

Let $Y = \log_{10}(y)$ and $X = \log_{10}(x)$. Then $Y = nX + C$, where $n$ is the slope and $C = \log_{10}(a)$ is the Y-intercept.

Enter (x,y) coordinates (one pair per line, x > 0, y > 0):

Exponential: $y = ab^x$ (Log-Linear Transformation, Base 10)

This relationship becomes linear when $\log_{10}(y)$ is plotted against $x$.

$$ \log_{10}(y) = (\log_{10}b) \cdot x + \log_{10}(a) $$

Let $Y = \log_{10}(y)$ and $X = x$. Then $Y = MX + C$, where $M = \log_{10}(b)$ is the slope and $C = \log_{10}(a)$ is the Y-intercept.

Enter (x,y) coordinates (one pair per line, y > 0):

Exponential (Natural): $y = ae^{kx}$ (Log-Linear Transformation, Natural Log)

This relationship becomes linear when $\ln(y)$ is plotted against $x$.

$$ \ln(y) = kx + \ln(a) $$

Let $Y = \ln(y)$ and $X = x$. Then $Y = kX + C$, where $k$ is the slope and $C = \ln(a)$ is the Y-intercept.

Enter (x,y) coordinates (one pair per line, y > 0):

Relevant Tools

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