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Rewrite the logarithm as a ratio of common logarithms and natural logarithms.

log2.8(x)
(a) common logarithms
X
(b) natural logarithms

Rewrite the logarithm as a ratio of common logarithms and natural logarithms. log-example-1

1 Answer

2 votes

Answer:


\textsf{a)} \quad (\log(x))/(\log(2.8))


\textsf{b)} \quad (\ln(x))/(\ln(2.8))

Explanation:

Part (a)

The common logarithm is the logarithm with base 10. It is often written as log without a specified base:


\log_(10)(x)=\log(x)

To rewrite
log_(2.8)(x) as a ratio of common logarithms, we can use the change of base formula.

The change of base formula is:


\large\boxed{\log_ba=(\log_na)/(\log_nb)}

where n is the new base.

In our case:

  • a = x
  • b = 2.8
  • n = 10

Substituting the values into the change of base formula gives:


\log_(2.8)(x)=(\log_(10)(x))/(\log_(10)(2.8))

As the common logarithm is often written as log without a specified base, this can also be written as:


\log_(2.8)(x)=(\log(x))/(\log(2.8))


\hrulefill

Part (b)

The natural logarithm is the logarithm with base e, where e is the Euler's constant approximately equal to 2.7182818 (7 d.p.). It is usually written as "ln":


\log_e(x)=\ln (x)

To rewrite
log_(2.8)(x) as a ratio of natural logarithms, we can use the change of base formula.

The change of base formula is:


\large\boxed{\log_ba=(\log_na)/(\log_nb)}

where n is the new base.

In our case:

  • a = x
  • b = 2.8
  • n = e

Substituting the values into the change of base formula gives:


\log_(2.8)(x)=(\log_(e)(x))/(\log_(e)(2.8))

As the natural logarithm is often written as "ln", this can also be written as:


\log_(2.8)(x)=(\ln(x))/(\ln(2.8))

User Mark Adams
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