Question

Solve the equation lg (4x + 5) = 0.8 Give your answer in three significant digits of accuracy. Answer: x=

Answer 1

Answer: 0.327

Step-by-step explanation: To solve an equation with logarithms, we have to remind three important features: 1) A logarithm is a function which only relates real positive numbers (zero is not included), but at certain cases it is possible to deal with negative numbers. 2) this function is an inverse function of any potential function and depends on the exponent and base values. 3) The logarithm function has some properties, which facilitates to solve any equation that includes such function at any of its terms. Considering the current case, the 2th and 3th feature will be useful to solve the equation, as follows.

Firstly, the logarithm is an inverse function of any potential
`10^(a)`(exponential) function and it is important to bear in mind the following definitions

`b^(a)=x, \\ \\log_a{x}=b`

The second definition is similar to the given equation of current the problem and therefore it is possible to do the following operation

`log_(10)(4x+5)=0.8`

Nevertheless, you are wondering why a sub index 10 is writting into the logarithm. When any logarithm is writing without any sub index is understood that the base is 10, which means that the inverse potential function has a base 10, for instance
`10^(a)`, where "a" stands for the exponent. Therein, at different literature in math the base 10 is omitted when a logarithm has no sub index, but it does not mean that the base 10 is disappeared, but it is not written for notation purposes.

Understanding the mentioned issued, we shall use the 2th and the 3th feature as follows,

`log_(10)(4x+5)=0.8\\\\10^{log_(10)(4x+5)}=10^(0.8)`

The previous step is useful because the "x" must be separated for having its value. At both sides of the equation is employed the base 10. Reducing terms, a new equation is written,

`10^{log_(10)(4x+5)}=10^(0.8)\\\\4x+5= 10^(0.8)\\\\x=(10^(0.8)-5)/(4)`

Doing the operations with the calculator the value of x is obtained,

`x=0.327393`

Considering the answer in three significant digits of accuracy means that it must count from the comma -left to right- at the third number and seeing the value of the fourth number. If the fourth number is greater than 5, then the third number must be approached to the highest number, otherwise remains equal. For the current case, the fourth number is 3, then

`x=0.327`

Remarks: what happen if another base into the logarithm is different of 10? The mentioned step / operation is applicable at any base.