Excel Logest Function

Related Function:

LINEST function

Equation for an Exponential Curve

The Excel Logest Function calculates the exponential curve that best fits a supplied set of y- and x- values.

If there is a single range of x-values, the calculated exponential curve satisfies the equation:

y = b * m^x

where,

x is the independent variable;
y is the dependent variable;
m is a constant base for the x value;
b is a constant which is the value of y when x = 0.

If there are multiple ranges of x-values, the calculated exponential curve satisfies the following equation:

y = b * m₁^x₁ * m₂^x₂ * ...

where,

the x's are the independent variable ranges;
y is the dependent variable;
the m's are constant bases for the x values;
b is a constant.

Function Description

The Excel LOGEST function returns statistical information on the exponential curve of best fit, through a supplied set of x- and y- values.

The basic statistical information returned is the array of constants, m_n, m_n-1, ... , b (or the constants m and b if there is a single range of x-values), for the exponential curve equation. However, you can also request that additional regression statistics be returned.

The syntax of the Logest function is:

LOGEST( known_y's, [known_x's], [const], [stats] )

where the function arguments are as follows:

known_y's

An array known y-values.

[known_x's]

An optional argument, providing an array of one or more sets of known x-values.

If provided the [known_x's] array should have the same length as the known_y's array;

If omitted, the [known_x's] array takes on the default value {1, 2, 3, ...}).

[const]

An optional logical argument that determines how the constant 'b' is treated in the exponential curve equation y = b*m^x. This argument can have the value TRUE or FALSE, meaning:

TRUE (or omitted)	-	the constant b is treated normally.
FALSE	-	the constant b is set to 1 in the equation y = b*m^x.

[stats]

An optional logical argument which specifies whether or not you want the function to return additional regression statistics on the calculated curve.

This argument can have the value TRUE or FALSE, meaning:

FALSE (or omitted)	-	Do NOT return additional regression statistics. I.e. just return the array of values m_n, m_n-1, ... , b.
TRUE	-	DO return additional regression statistics. I.e. Return the array of values m_n, m_n-1, ... , b and also return the additional regression statistics listed in the table below.

The array of statistics returned from the Excel Logest function has the following form:

m_n	m_n-1	...	m₁	b
se_n	se_n-1	...	se₁	se_b
r₂	se_y
F	d_f
ss_reg	ss_resid

where the statistics returned are:

m_i	-	The array of constant base coefficients for the exponential curve equation
b	-	The constant value of y when x=0
se_i	-	The standard error values for the coefficients, m_i
se_b	-	the standard error value for the constant b (returns the #N/A error if the [const] argument is FALSE)
r₂	-	The coefficient of determination
se_y	-	The standard error for the y estimate
F	-	The F statistic, or the F-observed value
d_f	-	The number of degrees of freedom
ss_reg	-	The regression sum of squares
ss_resid	-	The residual sum of squares

Inputting an Array Formula

To input an array formula, you need to first highlight the range of cells for the function result. Type your function into the first cell of the range, and press CTRL-SHIFT-Enter.

Go to the Excel array formulas page for more details.

As the Logest function returns an array of values, it must be entered as an array formula. If the function is not entered as an array formula, only the first 'm' value in the calculated array of statistical information is returned.

You can see if a function has been input as an array formula, as curly brackets will be inserted around the formula, as it is viewed in the formula bar. This can be seen in the examples below.

Logest Function Example 1

Cells A2-A10 and B2-B10 of the spreadsheet below list a number of known x and known y values, and also shows these points, plotted on a chart. Cells D1-E5 of the spreadsheet show the results of the Excel Logest function, which has been used to return statistical information relating to the exponential curve of best fit through these points.

As shown in the formula bar, the formula for the Logest function is:

=LOGEST( B2:B10, A2:A10, TRUE, TRUE )

The curly brackets around this function show that it has been entered as an array formula.

Example of use of the Excel Logest Function

Cells D1 and E1 give the values of the base, m as 1.482939831, and the y-intercept, b as 2.257475168. Therefore, the equation for the exponential curve of best fit through the given points is:

y = 2.257475168 * 1.482939831^x

The remaining cells in the range D1-E5 give the following additional statistics for this curve:

The standard error value for the base m is 0.014718308
The standard error value for the constant b is 0.070073164
The coefficient of determination is 0.990327432
The standard error for the y estimate is 0.114007527
The F statistic is 716.6960934
The number of degrees of freedom is 7
The regression sum of squares is 9.315412472
The residual sum of squares is 0.090984014

Logest Function Example 2

Cells A2-A11, B2-B11 and C2-C11 of the spreadsheet below contain three different sets of independent variables (known x values), and cells D2-D11 of the spreadsheet contain the associated known y-values. Cells F1-H3 of the spreadsheet show the results of the Excel Logest function, which has been used to return statistical information relating to the exponential curve of best fit through these points.

As shown in the formula bar, the formula for the Logest function in this case is:

=LOGEST( D2:D11, A2:C11, TRUE, TRUE )

It is also seen, from the surrounding curly brackets, that the function has been entered as an array formula.

Example of use of the Excel Logest Function with multiple independant variables

Cells F1-I1 give the values of the coefficents, m₃, m₂ and m₁ as 2.010750937, 0.942167056 and 1.31373656, respectively and the y-intercept, b as 2.554652779. Therefore, the equation for the exponential curve of best fit through the given points is:

y = 2.554652779 * 1.31373656^x₁ * 0.942167056^x₂ * 2.010750937^x₃

The remaining cells in the range F1-I5 give the following additional statistics for this curve:

The standard error values for the coefficients m₃, m₂ and m₁ are 0.080315977, 0.012928031 and 0.04764347, respectively
The standard error value for the constant b is 0.275195565
The coefficient of determination is 0.997749047
The standard error for the y estimate is 0.057701675
The F statistic is 886.5125548
The number of degrees of freedom is 6
The regression sum of squares is 8.854886129
The residual sum of squares is 0.0199769

and the unused cells show the #N/A error.

For further information and examples of the Excel Logest function, see the Microsoft Office website.

Logest Function Errors

If you get an error from the Excel Logest function this is likely to be one of the following:

Common Errors

#REF!

Occurs if the array of [known_x's] is not the same length as the array of known_y's.

#VALUE!

Occurs if either:

Any of the values in the supplied [known_x's] or known_y's arrays are non-numeric (this may include text representations of numbers, as the Logest function does not recognise these as numeric values)

Either of the [const] or [stats] arguments can not be evaluated to TRUE or FALSE.

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